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REESE  LIBRARY' 


UNIVERSITY  OF  CALIFORNIA 

J&-V-  ,i8(>. 


Accession  No. 


O'.as  No. 


— U U U1 


.-  i        :  .  m  s      i 
I  I  •.  I  m  m 

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THE 


RESISTANCE  AND  PROPULSION 


OF 


SHIPS. 


BY 


WILLIAM   F.  DURAND, 

M 

Principal  of  the  School  of  Marine  Construction, 
Cornell  University. 


FJKST    EDITION. 
FIRST   THOUSAND. 


NEW   YORK: 

JOHN   WILEY   &   SONS. 
LONDON:   CHAPMAN    &   HALL,    LIMITED. 

1898. 


Copyright,  1898, 

BY 

WILLIAM    F     DURAND. 


ROBERT    IJKUMMOND,   ELKCTKOTYPRK   AND   I'RINTER.    NEW  YORK. 


PREFACE. 


DURING  the  last  twenty  or  thirty  years  the  literature 
relating  to  the  Resistance  and  Propulsion  of  ships  has 
received  many  valuable  and  important  additions.  Of  the  few 
books  in  English  published  in  this  period,  those  of  the  highest 
value  have  been  restricted  either  in  scope  or  in  mode  of 
treatment.  For  the  most  part,  however,  the  important  addi- 
tions to  the  subject  have  been  published  only  in  the  transac- 
tions of  engineering  and  scientific  societies,  or  in  the  technical 
press.  Such  papers  and  special  articles  are  far  from  provid- 
ing a  connected  account  of  the  trend  of  modern  thought  and 
practice,  and  the  present  work  has  been  undertaken  in  the 
hope  that  there  might  be  afield  of  usefulness  for  a  connected 
and  fairly  comprehensive  exposition  of  the  subject  from  the 
modern  scientific  and  engineering  standpoint. 

Such  a  work  must  depend  in  large  measure  on  the  extant 
literature  of  the  subject,  and  the  author  would  here  express 
his  general  acknowledgments  to  those  who  have  preceded 
him  in  this  field.  In  many  places  special  obligations  are  clue, 
and  it  has  been  the  intention  to  give  special  references  at 
such  points  to  the  original  papers  or  sources.  With  the 
material  drawn  from  the  general  literature  of  the  subject 
there  has  been  combined  a  considerable  amount  of  original 


IV  PREFA  CE. 

matter  which  it  is  hoped  will  contribute  somewhat  to  what- 
ever of  interest  or  value  the  work  as  a  whole  may  possess. 

A  free  use  has  been  made  of  calculus  and  mechanics  in  the 
development  of  the  subject,  the  nature  of  the  treatment 
requiring  the  use  of  these  powerful  auxiliaries.  At  the  same 
time  most  of  the  important  results  and  considerations  bearing 
on  them  are  discussed  in  general  terms  and  from  the  descrip- 
tive standpoint,  and  all  operations  involved  in  the  actual 
solution  of  problems  are  reduced  to  simple  expression  in 
terms  of  elementary  mathematical  processes.  It  will  thus  be 
possible,  by  the  omission  of  the  parts  involving  higher 
mathematics,  to  still  obtain  a  fairly  connected  idea  of  most 
of  the  subject  from  the  descriptive  standpoint,  and  to  apply 
all  methods  developed  for  purposes  of  design. 

In  the  development  of  such  methods  the  purpose  has 
been  to  supply  plain  paths  along  which  the  student  may  pro- 
ceed step  by  step  from  the  initial  conditions  to  the  desired 
results,  and  to  arrange  the  method  in  such  a  way  as  shall  con- 
duce to  the  most  intelligent  application  of  engineering  judg- 
ment and  experience.  In  the  design  of  screw-propellers 
especially,  where  the  number  of  controlling  conditions  is 
necessarily  large,  the  purpose  has  been  to  present  a  mode  of 
solution  in  which  the  most  important  controlling  conditions 
are  represented  in  the  formulae,  and  in  which  the  determina- 
tion of  the  numerical  values  of  these  representatives  by 
auxiliary  computation,  estimate,  or  assumption  is  forced  upon 
the  attention  of  the  student  in  such  a  way  as  to  call  at  each 
step  for  a  definite  act  of  engineering  judgment. 

In  so  far  as  the  method  may  differ  from  others,  it  is 
intended  to  present  the  series  of  operations  in  such  way  as  to 
favor  the  recognition  of  a  considerable  number  of  relatively 


PREFACE.  V 

simple  steps,  and  the  full  and  free  application  at  each  step  of 
such  judgment  and  precedent  as  may  be  at  hand.  Such 
methods  are  perhaps  better  adapted  to  the  discipline  of  the 
student  than  to  the  uses  of  the  trained  designer,  and  it  is 
obviously  for  the  former  rather  than  for  the  latter  that  such  a 
work  should  be  prepared.  At  the  same  time  it  seems  not 
unlikely  that  even  for  the  trained  designer  the  splitting  up  of 
concrete  judgments  into  their  separate  factors  is  often  an 
operation  of  value,  and  one  which  will  the  more  readily 
enable  him  to  adapt  his  methods  to  rapidly  changing  prece- 
dents and  conditions  of  design. 

The  natural  limitations  of  size  and  the  need  of  homo- 
geneity have  rendered  the  work  in  many  respects  less  com- 
plete than  the  Author  might  have  wished,  and  many 
important  developments  especially  in  pure  theory  have 
received  but  scanty  notice.  As  presented,  the  work  repre- 
sents substantially  the  lectures  on  Resistance  and  Propulsion 
given  by  the  Author  to  students  of  Cornell  University  in  the 
School  of  Marine  Construction,  and  many  features  both  in 
subject-matter  and  mode  of  treatment  have  been  introduced 
as  a  result  of  the  experience  thus  obtained  in  dealing  with 
these  subjects. 

CORNELL  UNIVERSITY,  ITHACA,  N.  Y., 
February  4,  1898. 


CONTENTS. 


CHAPTER   I. 
RESISTANCE. 

SECTION  PAGE 

1.  General  Ideas i 

2.  Stream-lines 7 

3.  Geometry  of  Stream  lines , 16 

4.  Khumt  of  General  Considerations , 28 

5.  Resistance  of  Deeply  Immersed  Planes  Moving  Normal  to  Themselves  32 

6.  Resistance   of   Deeply   Immersed    Planes  Moving   Obliquely  to   their 

Normal 37 

7.  Tangential  or  Skin  Resistance 40 

8.  Skin-resistance  of  Ship-formed  Bodies 49 

9.  Actual  Values  of  the  Quantities  /and  w  for  Skin-resistance 52 

10.  Waves 56 

11.  Wave-formation  Due  to  the  Motion  of  a  Ship-formed  Body  through  the 

Water. ...  77 

12.  The  Propagation  of  a  Train  of  Waves 89 

13.  Formation  of  the  Echoes  in  the  Transverse  and  Divergent  Systems  of 

Waves 91 

14.  Wave-making  Resistance 95 

15.  Relation  of  Resistance  in  General  to  the  Density  of  the  Liquid 106 

1 6.  Froude's  Experiments  on  Four  Models 106 

17.  Modification  of  Resistance  Due  to  Irregular  Movement 108 

18.  Variation  of  Resistance  due  to  Rough  Water 109 

19.  Increase  of  Resistance  due   to  Shallow  Water  or  to  the  Influence  of 

Banks  and  Shoals no 

20.  Increase  of  Resistance  due  to  Slope  of  Currents 119 

21.  Influence  on  Resistance  due  to  Changes  of  Trim 120 

22.  Influence  of  Bilge-keels  on  Resistance 123 

23.  Air-resistance 125 

24.  Influence  of  Foul  Bottom  on  Resistance 127 

25.  Speed  at  which  Resistance  Begins  to  Rapidly  Increase 127 

26.  The  Law  of  Comparison  or  of  Kinematic  Similitude 128 

27.  Applications  of  the  Law  of  Comparison 142 

vii 


Vlll  CONTENTS. 


28.  General  Remarks  on  the  Theories  of  Resistance 145 

29.  Actual  Formulae  for  Resistance 146 

30.  Experimental  Methods  of  Determining  the  Resistance  of  Ship-formed 

Bodies 148 

31.  Oblique  Resistance 151 

CHAPTER  II. 
PROPULSION. 

32.  General  Statement  of  the  Problem 157 

33.  Action  of  a  Propulsive  Element 159 

34.  Definitions  Relating  to  Screw  Propellers 169 

35.  Propulsive  Action  of  the  Element  of  a  Screw  Propeller 172 

36.  Propulsive  Action  of  the  Entire  Propeller 177 

37.  Action  of  a  Screw  Propeller  viewed  from  the  Standpoint  of  the  Water 

Acted  on  and  the  Acceleration  Imparted 195 

38.  ;The  Paddle-wheel  Treated  from  the  Standpoint  of  §  37 198 

39>   Hydraulic  Propulsion. »  203 

40.  Screw  Turbines  or  Screw  Propellers  with  Guide-blades 205 

CHAPTER   III. 
REACTION  BETWEEN  SHIP  AND  PROPELLER. 

41.  The  Constitution  of  the  Wake , 208 

42. /Definitions  of  Different  Kinds  of  Slip,  of  Mean  Slip,  and  of  Mean  Pitch  211 

43.  Influence  of  Obliquity  of  Stream  and  of  Shaft  on  the  Action  of  a  Screw 

Propeller 220 

44.  Effect  of  the  Wake  and  its  Variability  on  the  Equations  of  §  36 224 

45.  Augmentation  of  Resistance  due  to  Action  of  Propeller 229 

46.  A  nalysis  of  Power  necessary  for  Propulsion 230 

47.  Indicated  Thrust 239 

48.  Negative  Apparent  Slip. .  ... 240 

CHAPTER   IV. 
PROPELLER  DESIGN. 

49.  Connection  of  Model  Experiments  with  Actual  Propellers 243 

50.  Problems  of  Propeller  Design 261 

51.  General  Suggestions  Relating  to  the  Choice  of  Values  in  the  Equations 

for  Propeller  Design,  and  to  the  Question  of  Number  and  Location 

of  Propellers 284 

52.  Special  Conditions  Affecting  the  Operation  of  Screw  Propellers 292 

53.  The  Direct  Application  of  the  Law  of  Comparison  to  Propeller  Design  300 

54.  The  Strength  of  Propeller-blades 303 

55.  Geometry  of  the  Screw  Propeller , 310 


CONTENTS. 


IX 


CHAPTER  V. 

POWERING  SHIPS, 

SECTION  PACK 

56.  Introductory 326 

57.  The  Computation  of  Wp  or  the  Work  Absorbed  by  the  Propeller ,  327 

58.  Engine  Friction 328 

59.  Power  Required  for  Auxiliaries  336 

60.  Illustrative  Example 338 

61.  Powering  by  the  Law  of  Comparison 340 

62.  The  Admiralty  Displacement  Coefficient 343 

63.  The  Admiralty  Midship-section  Coefficient 347 

64.  Formulae  Involving  Welted  Surface 347 

65.  Other  Special  Constants 349 

66.  The  Law  of  Comparison  when  the  Ships  are  Not  Exactly  Similar 350 

67.  English's  Mode  of  Comparison 355 

68.  The  Various  Sections  of  a  Displacement  Speed  Power  Surface 357 

69.  Application  of  the  Law  of  Comparison  to  Show  the  General  Relation 

Between  Size  and  Carrying  Capacity  for  a  Given  Speed 359 

CHAPTER  VI. 

TRIAL    TRIPS. 

70.  Introductory 365 

71.  Detailed  Consideration  of  the  Observations  to  be  Made 367 

72.  Elimination  of  Tidal  Influence 371 

73.  Speed  Trials  for  the  Purpose  of  Obtaining  a  Continuous  Relation  be- 

tween Speed,  Revolutions,  and  Power 375 

74.  Relation  Between  Speed  Power  and  Revolutions  for  a  Long  distance 

Trial 381 

75.  Long-ccurse  Trial  with  Standardized  Screw 383 

76.  The  Influence  of  Acceleration  and  Retardation  on  Trial-trip  Data 386 

77.  The  Time  and  Distance  Required  to  Effect  a  Change  of  Speed 387 

78.  The  Geometrical  Analysis  of  Trial  Data 397 

79.  Application  of  Logarithmic  Cross-section  Paper 407 

80.  Steamship  and  Propeller  Data 414 


INTRODUCTORY  NOTE  RELATING  TO  UNITS  OF 
MEASUREMENT. 

THE  units  of  weight,  and  of  force  in  general,  used  in 
naval  architecture  are  the  pound  and  the  ton,  the  latter 
usually  of  2240  Ibs.  It  will  always  be  so  understood  in  the 
present  volume. 

The  units  of  velocity  are  the  foot  per  second,  the  foot  per 
minute,  and  the  knot.  The  latter,  while  often  used  in  the 
sense  of  a  distance,  is  really  a  speed  or  velocity.  As 
adopted  by  the  U.  S.  Navy  Department  it  is  a  speed  of 
6080.27  ft.  per  hour.  The  British  Admiralty  knot  is  a  speed 
of  6080  ft.  per  hour.  The  distance  6080.27  ft.  is  the  length 
of  a  minute  of  arc  on  a  sphere  whose  area  equals  that  of  the 
earth.  For  all  purposes  with  which  we  are  concerned  in  the 
present  volume  the  U.  S.  and  British  Admiralty  knots  may 
be  considered  the  same. 

On  the  inland  waters  of  the  United  States  the  statute 
mile  of  5280  feet  is  frequently  employed  in  the  measurement 
of  speed  instead  of  the  knot,  and  the  short  or  legal  ton  of 
2000  Ibs.  for  the  measurement  of  displacement  and  weight  in 
general. 

Revolutions  of  engines  are  usually  referred  to  the  minute 
as  unit. 


RESISTANCE    AND    PROPULSION    OF    SHIPS. 


CHAPTER    I. 
RESISTANCE. 

1.  GENERAL  IDEAS. 

IN  the  science  of  hydrostatics  it  is  shown,  for  a  body 
wholly  or  partially  immersed  in  a  liquid  and  at  rest  relative 
to  such  liquid,  that  the  horizontal  resultant  of  all  forces 
between  the  liquid  and  the  body  is  zero,  and  that  the  vertical 
resultant  equals  the  weight  of  the  body.  If,  however,  there 
is  relative  motion  between  the  liquid  and  the  body,  the 
hydrostatic  conditions  of  equilibrium  no  longer  hold,  and  we 
find  in  general  a  force  acting  between  the  liquid  and  the  body 
in  such  direction  as  to  oppose  the  movement,  and  thus  tend 
to  reduce  the  relative  velocity  to  zero.  This  force  which  we 
now  consider  is  therefore  one  called  into  existence  by  the 
motion.  It  will  in  general  have  both  a  horizontal  and  a 
vertical  component.  The  latter,  while  generally  omitted 
from  consideration,  may  in  special  cases  reach  an  amount 
requiring  recognition.  The  existence  of  such  a  resisting 
force,  while  due  ultimately  to  the  relative  motion,  may  be 
considered  more  immediately  as  arising  from  a  change  in  the 
amount  and  distribution  of  the  surface  forces  acting  on  the 


2  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

body.  When  there  is  no  relative  motion  between  body  and 
liquid,  the  surface  forces  are  wholly  normal  pressures  and 
their  distribution  is  such  that,  as  above  noted,  the  horizontal 
resultant  is  zero  and  the  vertical  resultant  equals  the  weight. 
The  instant  such  motion  arises,  however,  the  surface  forces 
undergo  marked  changes  in  both  amount  and  character.  The 
normal  pressures  are  more  or  less  changed  in  amount  and  dis- 
tribution, and,  in  addition,  tangential  forces  which  were 
entirely  absent  when  the  body  was  at  rest  are  now  called  into 
existence.  In  consequence  the  horizontal  resultant  is  no 
longer  zero,  but  a  certain  amount  R,  the  equal  of  which  must 
be  constantly  applied  in  the  direction  of  motion  if  uniform 
movement  under  the  new  conditions  is  to  be  maintained. 
The  entire  vertical  resultant  must,  of  course,  still  equal  the 
weight.  This  resultant,  however,  may  be  considered  as  made 
up  of  two  parts,  one  due  to  the  motion,  and  the  other  to  the 
statical  buoyancy  of  the  portion  immersed.  The  sum  of 
these  two  will  equal  the  weight.  Hence  when  in  motion  the 
statical  buoyancy  will  not  in  general  be  the  same  either  in 
amount  or  distribution  as  for  the  condition  of  rest. 

The  amount  and  distribution  of  the  surface  forces  must 
depend  ultimately  on  the  following  conditions: 

f  (a)  Geometrical  form    and    dimensions 

(i)  The  body -j  of  immersed  portion. 

[_  (b)  Character  of  wetted  surface. 

(a)  Density. 

(b)  Viscosity. 
(3)  The  relative  motion. 


(2)  The  liquid...  j 


Since  it  is  the  relative   velocity   between  the  body  and 
liquid  with  which  we  are  concerned,  it  is  evident  that  we  may 


RESISTANCE.  3 

approach  the  problem  from  two  standpoints  according  as  we 
consider  the  liquid  at  rest  and  the  body  moving  through  it, 
or  the  body  at  rest  and  the  liquid  flowing  past  it.  The 
former  is  known  as  Euler's  method,  and  the  latter  as  La- 
grange's.  Each  method  has  certain  advantages,  and  it  will 
be  found  useful  to  view  the  phenomena  in  part,  at  least, 
from  both  standpoints. 

We  may  also  view  the  constitution  of  the  resisting  force 
in  two  ways.  The  first,  as  a  summation  of  varying  forces 
acting  between  the  water  and  the  elements  of  the  surface  as 
mentioned  above.  On  the  other  hand  considering  the  water, 
we  find  as  the  result  of  this  disturbance  in  the  distribution 
and  amount  of  the  liquid  forces  a  certain  series  of  phenomena 
varying  somewhat  with  the  location  of  the  body  relative  to 
the  surface  of  the  liquid.  These  we  will  briefly  examine. 

As  the  first  and  simplest  case  we  will  suppose  the  body 
wholly  immersed  and  so  far  below  the  surface  that  the  dis- 
turbances in  hydrostatic  pressure  become  inappreciable  near 
the  surface.  It  thus  results  that  there  are  no  surface  effects 
or  changes  of  level. 

Let  ABKj  Fig.  I,  represent  the  body  in  question,  con- 
sidered at  rest  with  the  liquid  flowing  by.  Now  considering 
the  motion  of  the  particles  of  water,  it  is  found  by  experience 
that  two  well-marked  types  of  motion  may  be  distinguished. 

The  particles  at  a  considerable  distance  from  the  body 
will  move  past  in  paths  indistinguishable  from  straight  lines. 
As  we  approach  the  body  we  shall  find  that  the  paths  become 
gently  curved  outward  around  the  body  as  shown  at  LMN. 
Such  paths  are  in  general  space  rather  than  plane  curves. 
The  nearer  we  approach  to  the  body  the  more  pronounced 
the  curvature,  but  in  all  these  p«aths  the  distinguishing  feature 


4  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

is  the  smooth,  easy-flowing  form  and  the  absence  of  anything 
approaching  doubling  or  looping.  Such  curves  are  known  as 
stream-lines.  Passing  in  still  nearer  the  body,  however,  we 
shall  find,  if  its  form  is  blunt  or  rounded,  a  series  of  large 
eddies  or  vortices  seemingly  formed  at  or  near  the  stern. 
These  float  away  and  involve  much  of  the  water  extending 
from  the  body  to  some  little  distance  sternward.  The  water 
between  the  eddies  will  also  be  found  to  be  moving  in  a  more 


FIG.  i. 

or  less  confused  and  irregular  manner.  These  eddies  with 
the  water  about  and  between  them  constitute  the  so-called 
"  wake."  At  high  speeds  also,  and  with  a  blunt  or  broad 
forward  end,  there  may  be  found  in  addition  just  forward  of 
the  body  a  small  mass  of  water  in  which  the  motion  is  not 
well  defined  and  smooth.  Again,  at  the  sides  and  between 
the  smooth-flowing  stream-lines  and  the  body  we  shall  find  a 
belt  of  confused  eddying  water  in  which  the  loops  and  spiral 
paths  are  very  small,  the  whole  constituting  a  relatively  thin 


RESISTANCE. 


5 


layer  of  water  thrown  into  the  most  violent  confusion  as 
regards  the  paths  of  its  particles.  The  extent  of  the  disturb- 
ance decreases  gradually  from  the  body  outward  to  the  water 
involved  in  the  smooth  stream-line  motion.  The  character- 
istic feature  of  the  motion  of  the  water  involved  in  these 
vortices,  eddies,  etc.,  is  therefore  its  irregularity  and  com- 
plexity as  distinguished  from  the  smoothness  and  simplicity 
of  that  first  considered.  The  volumes  KEF  and  ABC  are 
sometimes  known  as  the  liquid  prow  and  stern.  The  former, 
however,  is  usually  inappreciable  in  comparison  with  the 
latter.  The  term  " dead-water  "  is  also  sometimes  applied  to 
the  mass  of  water  thus  involved. 

We  may  also  consider  the  path  of  the  particles  by  Ruler's 
method  of  investigation.  This  will  be  simply  the  motion  of 
the  particle  relative  to  the  surrounding  body  of  water  consid- 
ered as  at  rest.  For  the  outlying  particles,  such  as  those 
forming  the  curves  LMN,  we  shall  have  now  simply  a  move- 
ment out  and  back  as  the  body  moves  by.  The  paths  out 
and  back  are  not  usually  the  same,  and  the  particle  does  not 
necessarily  or  usually  return  to  its  starting-point.  If  we  now 
suppose  the  body  moving  toward  G,  the  path  of  a  particle 
originally  at  L  will  be  some  such  curve  as  LRS.  This  is  a 
single  definite  path,  the  particle  being  moved  out  from  L  and 
finally  brought  to  5  without  final  velocity,  where  it  therefore 
remains.  For  particles  involved  in  the  eddies  and  whirls 
however,  the  paths  are  entirely  indefinite,  and  consist  of 
confused  interlacing  spirals  and  loops.  The  particle  is  not 
definitely  taken  hold  of  at  one  point  and  as  definitely  left  at 
another,  but  instead  may  be  carried  with  the  body  some 
distance  and  then  left  with  a  certain  velocity  and  energy,  the 
latter  gradually  and  ultimately  becoming  dissipated  as  heat. 


O  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

Let  us  now  remove  the  special  restriction  relating  to  the 
location  of  the  body,  and  let  us  suppose  the  motion  to  take 
place  at  or  near  the  surface  of  the  liquid.  The  normal  dis- 
tribution of  hydrostatic  pressure  near  the  surface  will  now  be 
disturbed,  and,  as  an  additional  result,  changes  of  elevation 
will  occur  constituting  certain  series  of  waves.  As  we  shall 
see  later,  the  energy  involved  in  these  waves  is  partly 

propagated  on  and   retained  within  the  system,   and   partly 

/ 
propagated  away  and  lost. 

Now  returning  to  the  general  consideration  of  resistance, 
it  is  evident  that  we  may  approach  its  estimation  from  two 
standpoints,  (i)  We  may  seek  to  study  the  amount  and 
nature  of  the  disturbance  in  the  distributed  liquid  forces  act- 
ing on  the  immersed  surface.  (2)  We  may  seek  to  know  the 
resistance  or  force  between  the  body  and  the  liquid,  through 
its  effects  on  the  latter  as  manifested  in  the  various  ways 
above  described.  The  latter  is  the  point  of  view  usually 
taken.  From  this  standpoint  it  seems  natural  to  charge  a 
portion  of  the  resistance  to  each  manifestation,  and  thus  to 
look  for  a  part  in  the  production  of  the  curved  stream-lines,  a 
part  in  the  production  of  the  eddies  at  the  bow  and  stern,  a 
part  in  the  eddying  belt  due  to  tangential  or  frictional  forces, 
and  a  part  in  the  waves. 

These  various  manifestations  we  shall  proceed  to  take  up 
in  order. 

Stream-lines  and  stream-line  motion  have  played  so 
prominent  a  part  in  the  various  views  which  have  been  held 
on  resistance,  and  serve  so  well  to  show  certain  features  of 
the  general  problem,  that  we  shall  find  it  profitable  to  first 
examine  them  in  some  detail. 


RESISTANCE. 


2.  STREAM-LINES. 

For  the  definition  and  general  description  of  what  is 
meant  by  a  stream-line  we  may  refer  to  the  preceding  section. 
We  have  now  to  define  a  stream-tube  or  tube  of  flow.  Let 
PQRS,  Fig.  2,  be  any  closed  curve  in  a  plane  perpendicular 


FIG.  2. 

to  the  line  of  motion  LK.  Let  this  curve  be  located  at  a 
point  so  far  from  AB  that  the  stream-lines  PUV>  etc.,  are  at 
/'sensibly  parallel  to  KL.  The  particles  comprised  in  the 
contour  PQRS  at  any  instant  will,  in  their  passage  past  AB, 
trace  out  a  series  of  paths  which  by  their  summation  will  form 
a  closed  tube  or  pipe.  Such  is  called  a  tube  of  flow.  From 
these  definitions,  a  particle  of  water  which  is  within  the  tube 
at  PR  will  always  remain  within  it,  and  no  others  will  enter. 
We  shall  therefore  have  a  tube  of  varying  section  in  which 
the  water  will  flow  as  though  its  walls  were  of  a  frictionless 
rigid  material,  instead  of  the  geometrical  boundary  specified. 
Starting  at  PR  with  a  certain  amount  of  water  filling  the 
cross-section  of  the  tube,  we  find  the  motion  parallel  to  KL. 
As  we  approach  and  pass  AB  the  direction  and  velocity  of 
flow  will  change,  but  the  latter  always  in  such  way  as  to 
maintain  the  tube  constantly  full.  After  passing  beyond  AB 


RESISTANCE  AND   PROPULSION  OF  SHIPS. 

the  direction  of  motion  will  approach  KL,  and  finally  at  a 
sufficient  distance  will  again  become  parallel  to  it.  Suppos- 
ing the  fundamental  conditions  to  remain  unchanged,  the 
entire  configuration  of  stream-lines  will  remain  constant,  and 
at  any  one  point  there  will  be  no  variation  from  one  moment 
to  the  next.  It  follows  that  the  conditions  for  steady 
motion  as  defined  in  hydrodynamics  are  fulfilled,  and  hence 
that  the  equations  of  such  motion  are  directly  applicable  to 
the  liquid  moving  within  the  tube.  Assuming  for  the 
present  that  the  liquid  is  hydrodynamically  perfect,  that  is, 
that  there  are  no  forces  due  to  viscosity,  the  equation  for 
steady  motion  is 


where  /  =  pressure  per  unit  area; 
cr  ==  density; 

z  =  elevation  above  a  fixed  datum  ; 
v  =  velocity; 

h  =  a  constant  for  each  stream-line,  called  the  total 
head. 

—  is'  called  the  presure-head  ; 
cr 

z  is  the  actual  head  ; 

v9 

—  is  called  the  velocity-head. 

The  total  head,  being  constant  for  each  line  or  indefinitely 
small  tube,  but  variable  from  one  to  another,  may  be  consid- 

*  We  shall  not  here  develop  the  fundamental  equations  of  hydro- 
dynamics, but  shall  assume  the  student  familiar  with  them  or  within  reach 
of  a  text-book  on  the  subject. 


RESISTANCE.  9 

ered  as  a  distinguishing  characteristic  of  the  tube  or  line  of 
flow.  If  we  take  the  value  of  such  characteristic  far  from 
AB,  as  at  PR,  we  shall  have  for  p  -=-  a  simply  the  statical 
pressure-head  measured  by  the  distance  from  PR  to  the  sur- 
face of  the  liquid.  For  s  we  have  the  vertical  distance  from 
PR  to  the  origin  O  at  any  fixed  depth.  The  sum  of  these 
two  equals  the  distance  from  O  to  the  surface.  Hence,  as  is 
perfectly  permissible,  if  we  take  O  at  the  surface,  /  -^  <j  -\-  z 
=  o,  and  h  =  V*  -r-  2gt  where  v0  is  the  known  velocity  with 
which  the  liquid  as  a  whole  is  moving  past  AB.  Hence  with 
O  at  the  surface  the  equation  to  a  line  becomes 


2g 


(2) 


It  is  evident  that  as  we  move  along  the  tube/  and  v  will 
depend  on  z  and  on  the  cross-sectional  area  of  the  tube.  If 
we  take  a  case  where  the  tube  is  sensibly  contained  in  a  hori- 
zontal plane,  z  will  be  constant  and  /  and  v  will  vary  in 
opposite  directions.  The  cross-sectional  area  will  be  found 
to  increase  somewhat  just  before  AB  is  reached.  Hence  in 
this  neighborhood  v  will  decrease  and  /  will  increase.  As  we 
pass  on,  the  area  will  decrease,  becoming  a  minimum  at  U. 
Here /will  be  a  minimum  and  v  a  maximum.  The  same 
changes  are  then  repeated  in  reverse  order  as  we  pass  on  from 
U  to  V.  The  reasons  for  the  variation  in  cross-sectional  area 
as  above  stated  may  be  seen  as  follows: 

Consider  a  tube  of  flow  of  large  diameter  to  entirely 
inclose  AB  and  the  liquid  about  it.  We  consider  the  sides 
of  the  tube  so  far  away  that  the  stream-lines  constituting  its 
boundary  are  sensibly  parallel  to  KL.  All  phenomena  may 
therefore  be  considered  as  taking  place  within  the  tube.  The 


IO 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


cross-sectional  area  available  for  the  flow  of  the  liquid  will 
evidently  be  the  area  taken  normal  to  the  stream-lines.  At 
the  entering  end  and  far  from  AB  this  will  be  a  right  section, 
as  MN.  Just  forward  of  AB,  however,  where  the  stream- 
lines curve  outward,  the  orthogonal  section  will  be  curved  or 
dished  as  indicated  by  HJ.  The  area  of  this  will  be  greater 
than  that  of  the  right  section  at  MN,  and  hence  the  average 
area  of  the  tubes  of  flow  must  be  greater.  This  effect  will 
also  be  more  pronounced  near  the  body,  where  the  change  in 
curvature  is  greater.  As  we  pass  on  to  £/the  stream-lines  are 
again  parallel  to  the  sides  of  the  large  tube,  and  hence  the 
net  section  available  for  flow  will  be  the  right  section  of  the 
tube  minus  the  section  of  AB.  Hence  the  average  cross- 
section  of  the  tubes  of  flow  must  be  less 'here  than  at  the 
entering  end.  As  above,  the  difference  will  be  more  pro- 
nounced near  the  body  than  far  removed  from  it.  Similarly 
just  behind  AB  the  orthogonal  section  will  be  curved  and  the 
average  area  will  be  increased,  as  at  HJ. 

We  have  now  to  show  that  in  any  tube  of  flow  in  which 
the  two  ends  are  equal  in  area,  opposite  in  direction,  and  in 
the  same  line,  the  total  effect  of  the  internal  pressures  is  o; 

that  is,  that  the  pressures  developed 
have  no  tendency  to  transport  the 
tube  in  any  direction  whatever. 

Let  us  first  consider  any  closed 
tube  or  pipe,  as  in  Fig.  3,  no  matter 
what  the  contour  or  the  variation  of 
sectional  area.  Suppose  it  filled 
with  a  perfect  or  non-viscous  liquid 
FlG-  3-  moving  with  no  tangential  force 

between  itself  and  the  walls   of    the  pipe.     The   conditions 


RESISTANCE. 


II 


are  therefore   similar  to    those  for  a  closed    tube    of    flow 

in  a  perfect    liquid.       Suppose  the  liquid    within    this    tube 

to  have  acquired  a  motion  of  flow  around  the  tube.     There 

being  no  forces  to  give  rise  to  a  dissipation   of  energy,    the 

motion  will  continue  indefinitely.      We  know  from  mechanics 

that    the    forces    in    such    case 

form    a    balanced    system    and 

hence  that  there  is  no  resultant 

force  tending  to  move  the  pipe 

in  any  direction.     That  such  is 

the   case  may  also  be  seen   by 

considering    that    if  there   were 

a    resultant     external    force    we 

might  obtain   work  by  allowing  FlG-  4- 

such  resultant  to  overcome  a  resistance.     Such  performance 

of    work  could  not   react  on  the  velocity  of   the  liquid,  and 

hence  we   should   obtain   work  without   the   expenditure   of 

energy. 

Suppose  next  the  pipe  to  be  circular  in  contour  and  of 
uniform  section,  as  in  Fig.  4. 

Let  a  =  area  of  section  of  pipe; 
r  =  mean  radius  of  contour; 
&  =  density  of  liquid  =  62.5  for  fresh  and  64  for  salt 

water; 
f  =  centrifugal  force  due  to  liquid. 

Then  the  volume  of  any  small  element  is  ardO,  and  the 
corresponding  centrifugal  force  is 


crardBv*        crav* 


12  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

Let  the  total  angle  A  OB  be  20,.  Let  6  be  the  angle 
between  OC  and  any  element.  Then  the  component  of  df 
along  OC  is 

<rav*  cos  BdB 


o 

Integrating,  we  have  for  the  total  resultant  along  OC 

20-av1 

Q=-       -  sin  0t. 
g 

This  total  force  outward  from  O  toward  C  may  be  equili- 
brated by  a  pair  of  tangential  tensions  at  A  and  B.  Let  each 
of  these  be  denoted  by  /.  Then  we  have 


x-k  , 

2p  sin  01  =  Q  -     sm  6lt 

o 


or 


That  is,  the  forces  due  to  the  movement  of  the  liquid  in 
any  arc  give  rise  to  two  tangential  tensions  at  the  ends  of  the 
arc,  and  these  tensions  are  independent  of  its  length  and 
depend  simply  on  a  and  v.  This  is  also  evidently  true  no 
matter  what  the  remainder  of  the  contour,  or  whether  it  is 
closed  or  not.  Hence  in  any  pipe,  closed  in  contour  or  not, 
containing  a  portion  of  uniform  area  and  curved  in  the  arc  of 
a  circle,  the  forces  due  to  the  flow  of  the  liquid  through  this 
portion  will  be  such  as  would  be  balanced  by  two  tangential 
tensions  at  the  ends  of  the  arc,  each  equal  to  crav*  -~  g. 

Let  us  now  consider  a  pipe  of  any  irregular  contour  and 
sectional  area,  as  in  Fig.  5.  Let  the  ends  A  and  D  be  of  the 
same  area,  but  turned  in  any  direction  relative  to  each  other. 


RESISTANCE.  13 

Suppose  liquid  as  above  to  flow  through  the  pipe.     Required 
to  determine  the  resultant  of  the  internal  pressures. 

Suppose  the  circuit  completed  by  means  of  a  pipe  AFED 
of  uniform  section  and  made  up  of  circular  arcs  AF  and  DE 
and  a  straight  length  FE.  Then  if  a  circulation  is  set  up  in 
this  closed  contour  such  that  the  velocities  at  A  and  D  are 
the  same  as  before,  the  conditions  in  ABCD  will  remain 
unaffected.  We  know  as  above  that  the  entire  resultant  is  o, 
and  that  the  effects  due  to  the  liquid  in  AF  and  DE  are 
represented  by  tensions  at  the  extremities  of  those  arcs  each 


FIG.  5. 

of  value  aav*  -±-  g.  The  effect  due  to  the  straight  part 
must  be  o,  and  the  two  tensions  at  F  and  E  will  balance  each 
other.  Hence  for  the  entire  resultant  of  the  forces  in  the 
dotted  part  of  the  contour  we  shall  have  the  equal  tangential 
forces  pl  =  AM  and  /3  =  DN  at  A  and  D.  Now  since  the 
system  of  forces  for  the  entire  contour  is  balanced,  it  follows 
that  the  forces  due  to  the  liquid  in  ABCD  must  be  repre- 
sented by  two  forces  equal  and  opposite  to  AM  and  DN. 
Hence  AR  and  DS  must  represent  in  direction,  point  of 
application,  and  amount  the  resultant  of  the  forces  due  to  the 
water  in  ABCD. 

As  a  special  case  let  ABCDE,  Fig.  6,  be  a  curved  pipe  of 
any  varying  sectional  area,  but  with  its  ends  A  and  E  equal, 


14  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

opposite  in  direction,  and  in  the  same  line.     Applying  the 
above  principle,  it  follows  that  the  resultant  will  be  that  of 


FIG.  6. 

the  two  forces  AR  and  ES,  equal  and  opposite  in  direction 
and  in  the  same  line.  Hence  in  such  case  the  resultant  is  o 
and  there  is  no  tendency  to  move  the  pipe  in  any  direction, 
and  in  particular  no  force  tending  to  move  it  in  the  direction 
AE.  This  serves  to  establish  the  initial  proposition  relative 
to  the  forces  in  such  a  tube  of  flow.  In  the  most  general 
case  the  straight  portions  at  the  two  ends,  while  parallel,  are 
not  in  the  same  straight  line.  The  preceding  treatment  still 
holds,  however,  and  we  shall  have  for  such  case  the  resultant 
represented  by  two  equal  and  opposite  forces  not  in  the  same 
line  and  therefore  forming  a  couple. 

Let  now  AB^  Fig.  7,  be  a  body  about  which  the  liquid 
flows  without   tangential  forces  in   stream-line  paths.     Let 


FIG.  7. 

this  body  be  immersed  at  an  indefinite  depth  so  that  surface 
effects  are  insensible.     We  wish  to  show  that  the  total  force 


communicated  to  the  body  by  the  stream-lines  is  o.  This 
may  be  seen  by  considering  PQRS  a  tube  of  flow  entirely 
surrounding  ACBD,  with  its  straight  ends  equal  in  area, 
opposite  in  direction,  and  in  the  same  line,  and  of  variable 
sectional  area,  ACBD  forming  an  internal  boundary.  The 
general  proposition  above  then  applies,  and  the  resultant  of 
all  forces  acting  on  the  internal  boundary  ACBD  will  be  o. 
Otherwise  the  tube  PQRS  may  be  considered  as  made  up  of 
an  indefinite  number  of  small  tubes  for  each  of  which  the 
resultant  may  be  a  couple.  It  is  easily  seen,  however,  that 
the  forces  constituting  these  couples  will,  for  the  whole  tube, 
constitute  a  system  uniformly  distributed  over  the  ends  PR 
and  QSy  and  hence  the  entire  resultant  will  be  o. 

This  important  conclusion  may  also  be  reached  by  con- 
sidering that  if  a  resultant  force  were  communicated  from  the 
liquid  to  AB,  we  might  obtain  work  by  allowing  AB  to  move 
and  thus  overcome  some  resistance.  By  the  supposition  of 
the  absence  of  all  tangential  force  between  AB  and  the 
liquid,  this  would  take  place  without  affecting  the  velocity  of 
the  latter.  In  fact  without  tangential  force  the  velocity  of 
the  liquid  can  in  no  wise  be  affected.  Hence  we  should 
obtain  work  without  the  expenditure  of  energy,  and  the  sup- 
positions leading  to  this  result  are  therefore  inadmissible. 

We  may  at  this  point  note  a  further  result  of  the  relation- 
ship expressed  by  the  general  equation  of  hydrodynamics 
above. 

Suppose  z  to  remain  substantially  constant  and  v  to 
increase  continually.  The  pressure  will  correspondingly 
decrease,  and  at  the  limit  for  a  certain  velocity  we  shall  have 
p  =  o.  An  indefinitely  small  increase  of  velocity  would  lead 
to  a  tendency  to  set  up  a  negative  pressure  or  tension  in  the 


1 6  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

liquid.  This  could  not  be  actually  realized,  and  the  result 
would  be,  instead,  a  breaking  up  of  the  stream  into  minute 
turbulent  whirls  and  eddies.  The  existence  of  such  turbu- 
lence may  therefore  imply  a  previous  condition  in  which 
there  existed  a  tendency  toward  an  indefinite  decrease  in  the 
pressure,  and  such  may  be  considered  as  its  hydrodynamical 
significance.  In  the  actual  case  the  stream  becomes  unsteady 
and  breaks  up  before  the  pressure  becomes  actually  reduced 
to  zero,  although  it  is  always  much  reduced  as  compared  with 
the  pressure  for  steady  flow.  A  further  result  of  the  great 
reduction  in  pressure  is  usually  found  in  the  giving  up  by  the 
water  of  the  air  which  it  holds  in  solution.  The  air  thus 
liberated  appears  in  the  form  of  small  bubbles  mingled  with 
the  water,  causing  the  foamy  or  yeasty  appearance  which 
usually  accompanies  this  phenomenon. 

3.  GEOMETRY  OF  STREAM-LINES. 

Among  the  first  to  note  the  relation  of  stream-lines  to  the 
problem  of  ship  resistance  was  Prof.  Rankine.  His  investi- 
gations covered  the  examination  of  their  general  properties 
and  the  derivation  of  forms  for  ships  which,  under  the  special 
conditions  assumed,  would  give  for  the  relative  motion  of 
ship  and  water  smooth  stream-line  paths.  We  will  now  show 
methods  of  constructing  stream-lines  for  various  special  cases 
as  developed  by  Rankine  and  other  investigators  since  his 
time. 

The  stream-lines  which  surround  a  body  are  in  general 
space-curves.  The  properties  of  such  lines  of  double  curva- 
ture are  so  complex  that  their  investigation  is  a  work  of  much 
difficulty  and  labor.  Many  helpful  and  instructive  sugges- 


RESISTANCE.  I/ 

tions,  however,  may  be  derived  from  a  study  of  plane  water- 
lines  to  which  the  present  notice  will  be  restricted. 

In  order  to  imagine  a  physical  condition  which  would  give 
such  lines,  suppose  a  body  partly  immersed  in  a  liquid  on  the 
surface  of  which  is  a  rigid  frictionless  plane.  Let  a  similar 
parallel  plane  touch  the  under  side  of  the  body.  Then  let 
the  liquid  between  the  planes  move  past  the  body,  or  vice 
versa  let  the  body  move  relative  to  the  liquid.  The  relative 
motion  of  liquid  and  body  may  then  be  considered  as  taking 
place  in  water-lines  contained  in  horizontal  planes.  At  the 
limit  we  may  suppose  the  planes  approached  very  near,  the 
body  being  then  simply  that  portion  contained  between  them. 
The  liquid  will  then  move  in  a  thin  horizontal  sheet  in  plane 
water-lines. 

It  is  shown  in  hydromechanics  that  for  such  water-lines 
there  exists  a  function  if>  such  that 


,     - 
u  =  velocity  along  x  =  — 


and 


-v  =  velocity  along  y  =  —  — ^-. 

ox 


That  is,  that  the  velocity  in  any  direction  is  the  rate  of 
change  of  this  function  in  a  direction  perpendicular  to  the 
given  one.  It  is  also  shown  that  this  function  is  constant  for 
any  one  water-line,  and  that  it  may  therefore  be  considered 
as  a  characteristic  or  equation  to  such  line.  It  is  likewise 
shown  that  its  value  for  any  one  line  is  proportional  to  the 
total  amount  of  flow  between  such  line  and  some  one  line 
taken  as  a  standard  or  line  of  reference. 


18 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


For  the  simplest  case  let  us  assume  two  planes  as  above, 
indefinite  in  extent  and  very  near.  At  a  given  point  let 
liquid  be  introduced  at  a  uniform  rate.  Such  a  point  of 
introduction  is  called  a  source.  In  such  case  the  liquid  will 
move  in  straight  lines  radiating  from  the  source.  Similarly 
we  might  have  assumed  liquid  abstracted  at  a  uniform  rate. 
Such  a  point  of  abstraction  is  called  a  sink.  In  such  case 

likewise,  the  liquid  will  move 
in  straight  lines  converging 
toward  the  sink. 

In  Fig.  8  let  O  be  the  point 
and  OA  the  line  of  reference 
from  which  the  total  flow  is 
measured.  Let  a  be  the  amount 
introduced  per  second  or  other 
unit  of  time,  and  B  the  angle 
FIG.  8.  between  OA  and  any  line  OP. 

Then  the  flow  between  OA  and  OP  is  that  corresponding  to 
the  angle  A  OP.     Hence  we  have 


at) 

ih  = . 

~          27T 


The  quantity  a  -r-  2n  is  called  the  strength  of  the  source  or 
sink. 

It  is  also  shown  in  hydromechanics  that  the  function  $  for 
a  complex  system  of  sinks  and  sources  is  simply  the  algebraic 
sum  of  the  separate  functions,  considering  i/>  for  a  source  as 
-|-  and  for  a  sink  as  — .  Hence  for  such  a  system  in  general 
we  have 


RESISTANCE. 

For  two  sources  of  equal  strength  we  have 


,  ^ 

.  —  -         and      t/-   =  —  . 

l  27T  T*          271 


Hence 


For  a  sink  and  source  of  equal  strength  we  have 


a 


In  all  such  cases  of  simple  combination  the  actual  stream- 
lines are  most  readily  found  by  a  construction  due  to  Maxwell 
and  illustrated  in  Fig.  9. 

From  O1  and  <9a  let  radiating  lines  be  drawn  at  angular 
intervals  proportional  to  a  -f-  2?r.  Then  the  number  of  lines 


will  also  be  proportional  to  a  H-  2n.  Take  any  point  P. 
Then  it  is  readily  seen  that  by  going  across  the  quadrilateral 
PQRS  to  R  we  find  another  point  for  which  ^  is  increased  by 


20  RESISTANCE   AND    PROPULSION   OF  SHIPS. 


RESISTANCE. 


21 


22  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

the  angle  RO^S  and  6^  decreased  by  the  equal  angle  QO^S. 
Hence  at  R  we  shall  have  (61  -f-  #2)  of  the  same  value  as  at  P. 
Hence  P  and  R  will  be  points  on  one  of  the  stream-lines 
belonging  to  the  two  sources  <9,  and  (9a.  It  follows  that  the 
series  of  stream-lines  for  these  sources  will  be  obtained  by 
drawing  continuous  curved  lines  diagonally  across  the  series 
of  quadrilaterals  formed  by  the  intersections  of  the  radiating 
lines  from  Ol  and  6>2.  Similarly  it  may  be  seen  that  for  Q 
and  5  the  values  of  (#t  —  #a)  are  the  same,  and  hence  that 
these  two  points  are  on  one  of  the  stream-lines  for  a  source 
Ol  and  a  sink  <9a.  In  like  manner,  then,  the  series  of  stream- 
lines for  such  a  doublet,  as  it  is  termed,  will  be  found  by 
drawing  continuous  curvilinear  diagonals  in  the  other  direc- 
tion across  the  same  set  of  quadrilaterals.  These  construc- 
tions are  illustrated  in  Figs.  10  and  1 1,  and  it  is  readily  seen 
in  the  latter  case  that  the  curves  are  arcs  of  circles. 

Maxwell  showed  in  general  that  for  all  cases  of  combination 
of  two  systems  of  sinks  and  sources  or  their  equivalents,  the 
resulting  system  of  stream-lines  could  be  found  geometrically 
by  laying  down  those  for  the  two  component  systems,  and 
then  taking  curved  lines  continuously  diagonal  to  the  series 
of  quadrilaterals  thus  formed.  In  this  way  the  result  of  a 
combination  of  any  number  of  sinks  and  sources  may  be 
found.  In  practice  the  labor  involved  is  very  great  when 
they  exceed  three  or  four  in  number. 

In  case  the  sinks  and  sources  are  not  all  of  the  same 
strength,  exactly  the  same  method  holds. 

We  have  now  to  show  as  a  special  case  the  result  of 
the  combination  of  a  source  and  a  continuous  stream  with 
straight  stream-lines.  The  latter  may  be  considered  as  a  part 
of  the  system  due  to  a  source  of  infinite  strength  situated  at 


RESISTANCE.  2$ 

an  infinite  distance  away.  The  geometrical  method  for  the 
determination  of  the  resulting  stream-line  system  is  exactly 
similar  to  the  preceding,  and  comes  under  the  general  rule  of 
procedure  as  stated  above.  It  is  illustrated  in  Fig.  12.  The 
straight  lines  radiating  from  O  are  replaced  each  by  a  curved 
line  as  shown.  Outside  of  this  system  of  curved  lines  spring- 
ing from  O  are  the  deflected  lines  of  the  uniformly  flowing 
stream.  It  is  readily  seen  that  the  line  ABC  separates  one 
of  these  systems  from  the  other;  also  that  there  is  no  flow 
whatever  across  this  line,  and  hence  that  it  separates  the 
liquid  introduced  by  the  source  O  from  that  originally  in  the 
stream.  It  is  also  evident  that  if,  instead  of  the  source  O 
and  its  system  of  stream-lines,  we  should  substitute  a  friction- 
less  surface  ABC  extending  away  indefinitely,  the  lines  of  the 
uniform  stream  would  have  exactly  the  same  form  which  they 
now  have  as  a  result  of  the  combination  of  the  source  and 
stream. 

Taking  similarly  a  uniform  stream  in  combination  with  a 
source  and  sink,  we  have  the  result  of  Fig.  13.  In  this  case 
it  is  seen  that  the  liquid  introduced  at  O,  and  withdrawn  at 
O^  does  not  spread  away  indefinitely,  but  is  confined  to  the 
portion  bounded  by  the  line  ABC,  and  that,  as  before,  this 
line  separates  the  liquid  thus  introduced  and  withdrawn,  from 
that  originally  in  the  stream.  Hence  also  if,  instead  of  the 
doublet  O.O^,  we  should  introduce  a  thin  flat  solid  with  fric- 
tionless  contour  ABC  .  .  .  ,  the  resulting  system  of  stream- 
lines in  the  uniform  stream  would  be  the  same  as  that  here 
resulting  from  the  stream  and  the  doublet. 

It  is  thus  seen  that  we  may  in  this  way  arrive  at  the 
system  of  stream-lines  due  to  a  frictionless  solid  of  certain 
form  plunged  in  a  uniformly  flowing  stream. 


24  RESISTANCE  AND    PROPULSION  OF  SHIPS. 


RESISTANCE. 


5      ! 


26 


RESISTANCE  AND   PROPULSION   OF  SHIPS. 


The  form  ABC  ...  is  not,  however,  suitable  for  the 
water-line  of  a  ship,  and  it  is  only  by  an  extension  of  the 
method  that  stream-lines  for  ship-shaped  forms  may  be 
determined.  This  extension  involves  the  appropriate  choice 
and  location  of  sources  and  sinks  such  that  the  resulting  line 
of  separation  shall  as  closely  as  may  be  desired  resemble  a 
ship's  water-line.  The  most  general  method  of  effecting  this 
extension  is  that  due  to  Mr.  D.  W.  Taylor,*  which  we  may 
briefly  summarize  as  follows: 

Instead  of  a  system  of  separate  sources  and  sinks, 
Mr.  Taylor  imagines  a  source-and-sink  line  or  narrow  slot 
through  a  part  of  which  liquid  is  introduced,  and  through  the 
remainder  of  which  it  is  withdrawn.  The  variation  of 
strength  along  this  line  corresponds  to  a  difference  in  width 
of  slot.  The  distribution  of  strength  may  be  represented 
graphically  by  a  line  ABC,  Fig.  14,  where  OB  represents 
the  source  portion  and  BP  the  sink  portion,  the  strength 
at  any  point  being  proportional  to  the  ordinate  of  ABC. 


FIG.  14. 

The  areas  ABO  and  BPC  must  be  equal,  else  the  total 
amounts  introduced  and  withdrawn  would  not  be  the  same. 
The  author  then  shows  by  graphical  representation  and  inte- 

*  Transactions  Institute  of  Naval  Architects,  vol.  xxxv.  p.  385. 


RESISTANCE. 


gration  how  to  find  the  current  function  or  equation  for  the 
stream-lines  corresponding  to  such  a  sink-and-source  line,  and 
how  by  varying  the  character  of  ABC,  Fig.  14,  to  vary  the 
character  of  the  resulting  contour  ABC  .  .  .  ,  Fig.  13,  so  as 
to  bring  it  to  any  desired  degree  of  similarity  to  a  given 
water-line. 

The  distribution  of  stream-lines  being  known,  it  is  a 
simple  matter  to  find,  by  the  help  of  the  equation  for  steady 
motion,  §  2  (i),  the  corresponding  distribution  of  pressure. 
This  is  illustrated  in  Fig.  15.  ABC 
represents  one  quarter  of  the  hori- 
zontal section  of  a  body  producing 
a  system  of  stream-lines  as  shown. 
Then  at  points  along  a  line  OY1 
the  excess  in  pressure  over  the 
normal  may  be  represented  by  the 
ordinate  to  the  curve  DE  relative 
to  O  Yl  as  axis.  Similarly  the  defect 
in  speed  at  the  same  points  may  be 
represented  by  a  curve  such  as  FG. 
In  like  manner  at  points  along  the  FlG-  I5- 

line  BY^  the  excess  of  speed  is  represented  by  the  curve  HI 
and  the  defect  of  pressure  byJK.  At  a  considerable  distance 
from  the  body  these  values  all  approach  indefinitely  near  to 
their  normal  values  and  the  variations  therefrom  become 
indefinitely  small,  as  indicated  by  the  diagram. 

Since,  under  the  special  conditions  assumed  in  the  treat- 
ment of  stream-lines,  the  total  resultant  force  between  the 
body  and  the  liquid  is  o,  the  resistance  to  the  motion  of  the 
solid  through  the  liquid  will  also  be  o.  The  investigation  of 
stream-lines  is  not  therefore  of  importance  in  the  determina- 


28  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

tion  of  the  actual  resistance  in  any  given  case,  but  rather  as 
showing,  under  the  assumptions  of  plane  stream-lines  and  no 
tangential  force  between  the  surface  and  the  liquid,  what 
forms  would  give  rise  to  smooth  and  easy  stream-lines,  and 
hence  presumably  in  an  actual  liquid  would  experience  the 
minimum  of  resistance. 

In  the  actual  case  the  liquid  is  quite  free  instead  of  being 
confined  between  two  rigid  planes.  This  introduces  into  the 
form  of  the  stream-lines,  especially  near  the  surface,  profound 
changes,  so  that  the  indications  received  from  plane  stream- 
lines must  be  used  with  caution,  and  only  so  far  as  compari- 
son with  the  results  of  experimental  investigation  may  seem 
to  justify.  See  further  §  14,  on  wave-resistance. 

4.  RESUME  OF  GENERAL  CONSIDERATIONS. 

In  considering  the  application  of  the  general  principles 
developed  in  the  present  chapter  to  the  resistance  of  ships,  it 
must  be  borne  in  mind  that  for  the  present  we  take  no 
account  of  any  influence  due  to  the  presence  of  a  propelling 
agent.  The  resistance  which  we  here  consider  is  the  actual 
or  tow-rope  resistance — the  resistance  which  the  ship  would 
oppose  to  being  moved  as  by  towing  through  the  water  at  the 
given  speed.  The  modifications  due  to  the  presence  of  a 
propeller  or  paddle-wheel  will  be  considered  in  §  45. 

We  have  before  us,  therefore,  the  general  problem  of  the 
resistance  of  a  partially  immersed  body  in  an  actual  liquid,  as 
water.  For  convenience  we  shall  examine  the  subject  under 
the  following  heads: 

(i)  Stream-line  Resistance. — The  liquid  being  no  longer 
without  viscosity  or  internal  shearing  stresses,  the  stream-line 
conditions  of  a  perfect  liquid  will  not  be  quite  realized,  and 


RESISTANCE. 


I 


the  maintenance  of  these  lines  in  the  actual  liquid  will  involve 
a  slight  drain  of  energy  from  the  body.  This  involves  the 
transmission  of  a  resultant  force  between  the  liquid  and  the 
body  opposed  to  the  relative  motion  of  the  two,  or,  in  the 
present  case,  to  the  motion  of  the  latter.  This  gives  the  first 
item  of  the  total  resistance  as  above. 

(2)  Eddy  Resistance. — We  come  next  to  the  water  involved 
in  the  liquid  prow  and  stern.     As  already  noted,  this  does 
not  follow  open  stream-line  paths.     It  is  thrown  into  irregular 
involved  motion,  the  energy  of  which  in  the  actual  liquid  is 
continually  degrading  into  heat.      The  water  involved  in  this 
motion  does  not,  moreover,  remain  the  same,  but  gradually 
changes  by  the  drawing  in  of  new  particles  and  the  expulsion 
of  others. 

The  maintenance  of  this  liquid  prow  and  stern  will  give 
rise,  therefore,  to  a  constant  drain  of  energy  from  the  moving 
body,  the  effect  of  which  is  manifested  as  a  resistance  to  the 
movement.  This  gives  the  second  item  as  above. 

(3)  Surface,    Skin,    or    Frictional   Resistance. — We    take 
next  the  belt  of  eddying  water  produced  by  the  action  of  the 
tangential  forces  between  the  surface  and  the  liquid.     These 
eddies  stand  in  the  same  relation  to  the  motion  of  the  body 
as  those  giving  rise  to  the  eddy  resistance  so  called.      Their 
maintenance  constantly  drains   energy  from    the   body,   and 
this  gives  rise  to  a  resistance  to  the  movement.     This  gives 
the  third  item  as  above. 

(4)  Wave  Resistance. — Finally  we  have  the  wave  systems 
which  always  accompany  the  motion  of  a  body  near  the  sur- 
face of  the  water.     As  remarked  in  §  I  and  as  we  shall  later 
show,  the  energy  involved  in  these  systems  is  not  retained 
intact,  but  a  portion  is  constantly  being  propagated  away  and 


3O  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

lost.  The  maintenance  of  the  wave  systems  requires,  there- 
fore, a  constant  supply  of  energy  from  the  moving  body,  and 
this,  as  in  the  previous  cases,  gives  rise  to  a  resistance  to  the 
motion.  This  gives  the  fourth  item  as  above. 

It  may  be  noted  that  in  the  most  general  view  (i)  and  (4) 
are  not  independent.  Waves  may  be  considered  as  the  result 
of  modified  stream-lines  near  the  surface,  and  hence  in  its 
most  general  significance  (i)  might  be  considered  as  including 
(i)  and  (4).  As  here  classified,  however,  we  mean  by  (i)  the 
resistance  due  to  the  maintenance  of  the  system  of  stream- 
lines which  would  be  formed  were  the  body  moved  with  the 
given  speed  at  an  indefinite  depth  below  the  surface.  The 
modification  due  to  wave-formation  is  then  taken  care  of 
by  (4).  In  any  case  (i)  is  very  small,  since,  with  the  veloci- 
ties occurring  in  practice,  water  behaves  nearly  as  a  non- 
viscous  liquid.  Similarly  (2)  and  (3)  might  with  propriety  be 
termed  the  eddy-resistance,  since  both  items  arise  from  the 
maintenance  of  systems  of  eddies.  Since  their  immediate 
causes  are  distinct,  however,  it  is  convenient  to  consider  them 
separately. 

The  sum  of  (i)  and  (2)  is  frequently  termed  the  head- 
resistance,  though  it  is  sometimes  considered  as  of  two  parts, 
called  head  and  tail  resistances,  the  former  being  due  to  the 
excess  of  pressure  in  the  liquid  prow,  and  the  latter  to  the 
defect  in  the  eddying  liquid  stern.  Inasmuch  as  they  cannot 
be  separated,  however,  we  shall  use  the  one  term  for  both. 

With  ships  of  ordinary  form  this  is  so  small  as  to  be 
relatively  negligible.  In  other  cases,  as  in  the  motion  of  the 
paddle-wheel,  propeller- blade,  bilge-keel  in  oscillation,  etc., 
this  is  one  of  the  chief  items  of  the  resistance  to  the  motion 
of  the  given  body. 


I 


RESISTANCE.  3 1 

In  ships  of  usual  form  we  are  therefore  concerned  chiefly 
with  surface  and  wave  resistance. 

The  sum  of  (i)  and  (4)  or  of  (i),  (2),  and  (4)  is  often 
known  as  the  residual  resistance.  This  term  has  reference  to 
the  usual  view  of  ship-resistance  which  recognizes  but  two 
chief  subdivisions: 

(a)  Surface  or  skin  resistance. 

(b)  Residual  resistance,  including  (4),  above,  and  all  other 
parts  not  properly  classified  under  (a). 

It  must  not  be  forgotten  that  this  classification  of  resist- 
ance according  to  the  various  effects  produced  on  the  water 
is  somewhat  arbitrary.  A  more  logical  mode  of  subdivision 
might  be  made  by  starting  from  the  body  and  considering 
that  each  element  of  immersed  surface  is  subjected  to  a  cer- 
tain force.  Such  force  may  be  resolved  into  two  com- 
ponents; one  tangential,  the  other  normal.  This  is  entirely 
general  and  includes  all  effects  no  matter  from  what  cause 
arising.  Comparing  with  the  other  method  of  subdivision  it 
is  evident  that  the  longitudinal  components  of  the  tangential 
forces  will  give  by  their  summation  the  surface  or  frictional 
resistance.  Hence  the  longitudinal  components  of  the  normal 
forces  must  give  by  their  summation  the  remaining  or  resid- 
ual resistance,  corresponding  to  (i),  (2),  and  (4),  or  with  ship- 
shaped  forms  almost  entirely  to  (4).  This  subdivision  is 
entirely  independent  of  any  supposition  with  regard  to  the 
effects  on  the  water,  and  is  the  most  direct  or  immediate  as- 
pect of  resistance  itself.  The  two  parts  may  be  termed 
tangential  and  normal  resistances. 

No  matter  what  view  we  may  take  of  the  subdivision  of 
resistance,  we  are  obliged  to  go  to  experimental  investigation 
for  all  satisfactory  information  as  to  its  amount.  We  pro- 


32  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

ceed,  therefore,  to  the  description  of  the  various  experimental 
investigations,  and  to  the  discussion  of  the  results  which  may 
be  derived  therefrom. 


5.  RESISTANCE   OF   DEEPLY  IMMERSED   PLANES   MOVING 
NORMAL  TO  THEMSELVES. 

If  the  body  in  Fig.  i  is  supposed  to  be  reduced  to  a  plane 
normal  to  the  line  of  motion,  the  result  will  be  a  system  of 
stream-lines  and  eddies  somewhat  as  in  Fig.  16,  though  the 


— o-g> 


FIG.  16. 

line  of  demarcation  between  the  two  is  not  fixed  nor  so  defi- 
nite as  must  be  suggested  by  a  rough  diagrammatic  represen- 
tation such  as  the  figure  gives.  The  tangential  resistance 
will  be  o  because  all  tangential  force  is  at  right  angles  to  the 
direction  of  motion.  This  does  not  mean  that  the  actual 
resistance  is  independent  of  the  character  of  the  surface,  but 
simply  that  the  actual  tangential  forces  have  no  longitudinal 
component.  The  nature  of  the  surface  will  presumably 
slightly  affect  the  stream-line  formation  and  the  amount  and 
nature  of  the  liquid  prow  and  stern,  so  that  ultimately  it  will 
have  its  effect  on  the  resistance.  Viewed  immediately, 
however,  the  resistance  is  due  wholly  to  the  eddies  and 
stream-lines  in  an  imperfect  liquid.  Experiment  shows  that 


RESISTANCE.  33 

the  resistance  in  such  case  may  be  approximately  expressed 
by  an  equation  of  the  form 


(i) 


where  R  =  resistance  in  pounds; 
f  =  a  coefficient; 
cr  =  density  of  liquid  —  62.5   for  fresh  and  64  for  salt 

water; 

g  =  acceleration  due  to  gravity  =  32.2  ft.  ; 
A  =  area  in  square  feet; 
v  =  velocity  in  feet  per  second. 

For  salt  water  <r  -f-  2g  is  very  commonly  put  equal  to  i. 
In  such  case  the  equation  becomes 


(2) 


The  values  of  the  coefficient  f  which  have  been  deter- 
mined experimentally  have  varied  widely.  This  is  doubtless 
due  to  the  fact  that  such  experiments  have  been  carried  on 
by  different  observers  under  different  conditions,  notably  as 
to  amount  of  surface,  depth  of  immersion,  and  actual  veloci- 
ties employed.  The  subject  has  not  been  examined  with 
sufficient  completeness  to  show  satisfactorily  the  general  rela- 
tion of  the  resistance  to  the  three  conditions,  area,  immer- 
sion, and  velocity.  In  fact  there  is  considerable  doubt  as  to 
whether  R  should  increase  strictly  as  the  area,  we  are  not  sure 
that  the  index  of  v  should  for  all  velocities  be  2,  and  the 
relation  of  /"to  varying  immersion  is  not  satisfactorily  known. 
The  attempt,  therefore,  to  express  the  results  of  experiments 
under  widely  varying  conditions  by  the  simple  formula  above, 
throwing  on  f  the  consequences  of  all  differences  between  the 


34  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

true  and  the  assumed  law,  may  with  justice  lead  to  widely 
varying  values  for  this  coefficient. 

Among  the  earliest  experiments  were  those  by  Col. 
Beaufoy  made  at  the  Greenland  Docks  in  London  between 
1793  and  1/98.  Taking  A  in  square  feet,  v  in  feet  per 
second,  and  R  in  pounds,  the  mean  value  of /deduced  from 
these  experiments  is  about  i.i. 

Mr.  Wm.  Froude's  experiments  on  planes  about  3  ft. 
wide  gave  with  similar  units  a  mean  value  of  about  1.7. 

Dubuat  at  a  single  speed  of  about  3.28  ft.  per  second 
found  for  a  plane  maintained  stationary  in  a  moving  stream 
a  value  of  f  =•  1.86.  For  a  stationary  liquid  and  moving 
plane,  the  speed  being  the  same,  he  found  f  =  1.43.  If  in 
the  first  case  the  liquid  moved  steadily  and  without  eddies  or 
internal  irregularities,  and  in  the  second  case  the  liquid  was 
strictly  at  rest,  the  relative  speed  being  the  same  in  each 
case,  these  two  results  should  be  the  same.  The  discrepancy 
was  doubtless  due  to  the  difficulty  of  making  satisfactory 
observations  on  the  mean  velocity  of  a  flowing  stream,  and 
also  to  the  now  well-known  difference  in  effect  between  a 
stream  moving  without  and  with  turbulence. 

More  recently  Joessel's  experiments  in  the  river  Loire, 
made  on  a  plane  of  sheet  iron  .98  ft.  high  by  1.31  ft.  long 
and  immersed  so  as  to  have  .66  ft.  of  water  over  the  upper 
edge,  gave  a  mean  value  of  f  =.  1.6.  The  maximum  velocity 
was  not  above  about  4.25  ft.  per  second. 

Still  more  recently  Mr.  R.  E.  Froude  has  determined  this 
coefficient  under  conditions  which  insure  the  highest  degree 
of  experimental  accuracy.  The  resulting  value  agrees  very 
closely  with  the  average  of  Beaufoy's  values  mentioned 
above,  and  was  about  i.i. 


RESISTANCE.  35 

An  unexplained  divergence  from  all  the  values  above 
given  is  indicated  by  the  results  of  experiments  on  bilge- 
keels.  An  attempt  to  fit  this  formula  of  resistance  to  these 
results  leads  to  a  coefficient  varying  from  5  or  6  to  15  or  16. 
Its  excessive  variability,  as  well  as  its  great  divergence  from 
the  other  values,  indicates  that  we  have  here  involved  some 
element  or  condition  not  represented  by  the  formula,  and 
whose  importance  has  been  hitherto  unsuspected.  It  is 
understood  that  experiments  under  the  direction  of  R.  E. 
Froude  are  in  progress  which,  it  is  expected,  will  throw  light 
on  these  points. 

It  may  be  well  here  to  note  that  if  the  immersion  is  suffi- 
ciently great  to  reduce  the  surface  disturbance  to  a  negligible 
amount,  the  resistance  will  be  practically  independent  of  any 
further  increase  of  depth.  This  is  readily  seen  from  the  fact 
that  the  resistance  arises,  not  from  the  pressure,  which  of 
course  increases  with  the  depth,  but  rather  from  a  difference 
in  the  distribution  of  such  pressure,  and  that  this  is  practi- 
cally independent  of  depth  so  long  as  surface  disturbance  is 
absent.  Still  otherwise  we  may  view  the  resistance  as  due  to 
the  energy  absorbed  by  the  stream-line  and  eddy  systems; 
and  since  below  the  minimum  depth  above  mentioned  the 
configuration  of  these  will  be  constant,  the  energy  absorbed 
and  the  resulting  resistance  will  likewise  remain  the  same. 
These  conclusions  are  borne  out  by  Beaufoy's  experiments. 

It  will  be  observed  that,  according  to  our  definition,  head- 
resistance  is  the  resistance  due  to  the  eddy  and  stream-line 
formation  supposing  the  immersion  of  the  body  so  great  that 
surface  effects  are  negligible.  It  is  not  the  resistance  due  to 
the  actual  stream-line  and  eddy  formation  if  near  the  surface, 
for  this  would  include  the  wave-resistance  as  well.  In  the 


30  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

present  case,  therefore,  tangential  resistance  being  absent,  we 
must  consider  the  change  in  the  total  resistance  as  the  plane 
approaches  the  surface  as  due  to  the  resistance  arising  from 
the  wave-formation.  In  any  experiment,  therefore,  if  the 
plane  is  near  the  surface,  the  resistance  actually  measured  will 
consist  of  two  parts:  head-resistance  proper  and  wave-making 
resistance. 

As  is  well  known,  a  very  slight  wave  may  involve  con- 
siderable energy,  especially  if  its  length  is  great.  Hence 
apparently  slight  surface  effects  may  count  for  much  in  the 
actual  total  resistance. 

It  seems  likely  that  the  explanation  of  much  of  the  varia- 
tion in  the  value  of  the  coefficient  f  noted  above  may  be 
found  in  these  considerations.  The  actual  amount  measured 
is  of  course  total  resistance.  If  the  immersion  is  not  sufficient 
to  eliminate  surface  disturbances,  then  this  total  resistance 
will  include  a  portion  due  to  the  wave-formation.  This  must 
depend  on  depth,  a  condition  not  represented  in  the  formula 
above.  As  we  shall  see  later  also,  wave-resistance  varies  with 
velocity  according  to  a  much  higher  index  than  2.  The 
formula  above  is  therefore  wholly  unsuited  to  represent  the 
wave-resistance  portion,  and  any  attempt  to  make  it  represent 
a  total  resistance  of  which  the  wave-making  portion  may  be 
relatively  important  will  naturally  lead  to  widely  divergent 
values  of  /.  In  the  case  of  bilge-keels  the  surface  disturb- 
ance is  of  very  considerable  amount,  and  it  seems  likely  that 
much  of  the  resistance  is  due  to  the  wave-making  effects 
produced.  The  attempt  to  fit  the  above  formula  to  the 
total  results  may  therefore  naturally  lead  to  a  value  of  / 
widely  differing  from  its  value  for  head-resistance  alone. 


RESISTANCE. 


37 


6.    RESISTANCE  OF  DEEPLY   IMMERSED   PLANES   MOVING 
OBLIQUELY  TO  THEIR  NORMAL. 

If  the  plane,  instead  of  standing  at  right  angles  to  the 
direction  of  motion,  is  inclined  at  an  angle  6,  we  have  a  result 
somewhat  as  indicated  in  Fig.  17.  In  such  case  the  force  in 


FIG.  17. 

the  line  of  the  normal  EB  as  determined  by  Joessel's  experi- 
ments is  given  by  the  formula 


P0=  1.622 


sin  0  cr 

.39  +  .61  sin  6  2g 


Atf: 


The  longitudinal  component,  or  the  resistance  in  the  line 
of  motion,  is  therefore 


Hence 


and 


R*  = 


P, 


sin2  8 


, 

.39  -|-  .61  sin  0  2g 


sn 


/>.~.39  +  -6i  sin  0 

RB_ sin'  6 

•^90    ~  -39  +  -61  sin  0" 


(4) 


*  Where  not  otherwise  explained,  the  nomenclature  of  this  section  is 
the  same  as  that  of  the  preceding. 


38  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

According  to    Beaufoy's    results   we    may  put    approxi- 
mately 


(5) 


and          7^-  =  sin8  8  .......     (6) 


90 


The  experiments  of  Wm.  Froude  also  agreed  in  giving 


It  must  be  admitted  that  the  condition  of  our  information 
on  this  subject  is  far  from  satisfactory,  and  further  experi- 
ments are  much  needed. 

Joessel's  experiments  were  also  directed  toward  the 
determination  of  the  location  of  the  center  of  the  system  of 
distributed  pressures  on  an  oblique  plane.  The  results  were 
as  follows: 

Let  /  be  the  length  of  plane  in  the  direction  of  motion, 
and  x  the  distance  to  the  center  of  pressure  from  the  leading 
edge.  Then 

*  =  (.195  +  .305  sin  #)/.       ....     (8) 

In  the  narrow  range  of  speeds,  o  to  4.25  ft.  per  second, 
this  law  seemed  to  be  independent  of  the  velocity. 

Referring  again  to  Fig.  17,  it  is  known  that  the  stream 
flowing  toward  AC  will  divide  and  pass  around,  a  part  by  C 
and  a  part  by  A.  Let  BK  be  the  plane  which  separates  one 
of  these  two  streams  from  the  other.  Then  Lord  Rayleigh 
has  shown  that  B  is  determined  by  the  following  proportion: 

AB  __  2  +  4  cos  8  —  2  cos3  0  -f  (n  —  B)  sin  B 
AC  ~  4+  n  sin  6 


RESISTANCE. 


39 


The  tangential  force  is  evidently  less  than  if  both  streams 
flowed  in  the  same  direction,  as  when  the  plane  is  moved 
parallel  to  itself.  Let  k  be  the  ratio  between  the  tangential 
forces  in  the  two  cases.  Then  Cotterill  *  has  shown  that 

4  cos  0  —  (n  —  20)  sin  0 
4  -\-  n  sin  0 

The  graphical  representation  of  these  relations  is  shown 
in  Fig.  1 8. 

The  importance  of  reducing  eddy  resistance  by  an  appro- 
priate choice  of  form  for  a  ship-shaped  body,  and  the  avoid- 


1.00 


9.  .so 
& 


*]  .60 


y  .50 


uJ  .40 
O 

o 


O  .20 
3 

i'10 


x 


10203040506070 
VALUES  OF  Q  IN  DEGREES 

FIG.    18. 


8090 


ance  where  possible  of  all  surfaces  normal  to  the  direction  of 
motion  and  of  all  sudden  changes  of  direction  in  the  surface, 
is  readily  seen.  It  thus  appears  as  shown  by  Taylor  f  for  a 

*  Transactions  of  the  Institute  of  Naval  Architects,  vol.  xx.  p.  152. 
•j-  "  Resistance  and  Screw  Propulsion,"  p.  27. 


4O  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

ship  300  feet  long  at  20  knots,  using  the  results  of  §  7  for 
tangential  resistance,  that  the  resistance  of  one  square  foot 
moving  normal  to  its  own  direction  is  about  770  times  as 
great  as  that  of  the  same  area  exposed  to  tangential  or  skin- 
resistance  only. 

We  may  also  add  in  this  connection  that  if  the  lines  of  a. 
ship  are  too  full  aft  for  her  speed,  large,  unstable  eddies  are 
liable  to  appear  about  the  stern,  suddenly  shifting  about  from 
one  quarter  to  the  other  and  causing  sudden  and  unsym- 
metrical  changes  in  the  total  resistance  and  in  the  action  of 
the  rudder,  thus  rendering  such  ships  very  difficult  to  steer. 


7.  TANGENTIAL  OR  SKIN  RESISTANCE. 

The  origin  of  this  form  of  resistance  has  been  referred  to 
in  §  I.  Experiments  for  the  determination  of  its  amount 
have  been  made  by  Beaufoy,  Tidman,  and  Wm.  Froude. 
Inasmuch  as  those  of  the  latter  are  the  most  recent  and  ex- 
tensive, we  shall  be  content  with  a  brief  resume  of  the  results. 

In  these  experiments  boards  T3¥  inch  thick,  19  inches 
deep,  and  of  different  lengths  from  4  to  50  feet  were  covered 
with  various  substances  and  towed  lengthwise  in  a  tank  of 
fresh  water,  their  speed  and  resistance  being  carefully  meas- 
ured by  appropriate  apparatus. 

For  the  discussion  of  the  results  of  the  experiments  the 
tangential  resistance  is  supposed  to  vary  according  to  the 
formula 


where  R  =  resistance  in  pounds; 
/=  a  coefficient; 


RESISTANCE. 


A  =  area  in  square  feet; 
v  =  velocity  in  feet  per  second  or  knots,  according  to 

the  value  of /used; 
n  =  an  exponent. 

A  summary  of  the  experimental  results  is  shown  in  the 
following  table: 

TABLE   I.    SHOWING    RESULTS   OF   EXPERIMENTS   ON 
SKIN-RESISTANCE. 


Nature  of 
Surface. 

Length  of  Surface  or  Distance  from  Cutwater. 

2  feet. 

8  feet. 

20  feet. 

50  feet. 

A 

B    |     C 

A 

B 

C 

A 

B 

C 

A 

B 

C 

Varnish  

2.00 

.41 

•390 

1.85 

.325 

.264 

1.85 

.278 

.240 

1.83 

.250 

.226 

Paraffin  

I.QC 

.38 

.770 

1.94 

.314     .260 

i-93 

.271 

.237 

Tinfoil  

2.16 

•30 

•  295 

1.99 

.2781   .263 

1.90 

.262 

.244 

1.83 

.246 

.232 

Calico  

1.93 

.8? 

.725 

1.92 

.626 

•  504 

1.89 

•531 

•447 

1.87 

•474 

.423 

Fine  sand.  .. 

2.00 

.81 

.690 

2.00 

.583 

.450 

2.OO 

.480 

•384 

2.06 

.405 

•337 

Medium  sand 

2.00 

.90 

•730 

2.00 

.625 

.488 

2.OO 

•  534 

.465 

2.00 

.488 

•456 

Coarse  sand  . 

2.00 

1.  10 

.880 

2.00 

.714 

.520 

2.OO 

.588 

.490 

Column  A  gives  the  value  of  n  for  the  particular  length 
and  character  of  surface. 

Column  B  gives  for  the  whole  surface  the  mean  resistance 
in  pounds  per  square  foot  at  a  speed  of  600  feet  per  minute 

Kr  10  feet  per  second. 
Column  C   gives  for  the   same  speed  the   resistance   per 
oquare  foot  at  a  distance  from  the  forward  end  equal  to  that 
stated  in  the  heading. 

It  is  thus  seen  that  the  resistance  per  unit  area  decreases 
as  we  go  from  forward  aft.  Thus  for  varnish  the  resistance 
per  square  foot  at  distances  of  2,  8,  20,  and  50  feet  from  the 
bow  are  respectively  .39,  .264,  .24,  .226.  It  necessarily 
results,  as  is  shown  by  the  table,  that  the  mean  resistance  per 
square  foot  decreases  with  increased  length.  The  explana- 
tion of  this  as  given  by  Mr.  Froude  is  as  follows: 


OF  THB 

UNIVERSITY 


42  RESISTANCE  AND   PROPULSION   OF  SHIPS. 

The  board  or  surface  under  test  moves  surrounded  by  a 
skin  of  eddying  water,  the  maintenance  of  the  energy  of 
which  gives  rise  to  the  resistance  under  consideration.  The 
forward  part  of  the  board  is  constantly  entering  water  at  rest,. 
while  the  after  portions  come  in  contact  with  water  already 
acted  on  by  the  forward  end,  and  possessing,  therefore,  to  a 
greater  or  less  extent,  this  eddying  motion.  In  these  eddies 
the  movement  of  the  water-particles  next  the  board  is 
forward,  more  or  less  of  the  water  being  involved,  and  with  a 
greater  or  less  velocity  according  to  the  length  of  time  they 
have  been  acted  on,  and  to  the  other  circumstances  of  the  sur- 
face and  motion.  It  follows  that  the  eddying  motion  will  be 
more  and  more  pronounced  as  we  go  from  the  bow  aft. 
Hence  a  part  of  the  board  near  the  forward  end  advancing 
into  still  water  will  naturally  meet  a  much  greater  resistance 
than  the  same  area  located  near  the  after  end,  and  entering 
water  which  already  possesses  in  large  measure  the  eddying 
motion. 

We  will  now  consider  in  detail  the  character  of  the  equa- 
tion for  tangential  resistance  as  given  above,  especially  as  to 
the  relation  between  the  quantities  involved  and  the  circum- 
stances of  the  experiment. 

The  quantity  A  is  taken  as  the  area  without  modification 
or  change.  It  is  seen  as  above  that  the  length  factor  of  area 
is  the  one  of  importance,  so  that  we  may  consider,  so  far  as 
its  effect  on  the  quantities  of  the  formula  is  concerned,  A  to 
be  represented  by  length.  Suppose  now  that  we  have  given 
a  series  of  values  of  R  for  a  range  of  values  of  quality,  length, 
and  speed.  Each  such  value  will  give  a  single  equation 
involving  the  two  unknowns /and  n.  These  may  be  both 
variable  from  point  to  point,  and  not  knowing  on  what  they 


RESISTANCE. 


43 


may  depend  we  cannot  consider  any  two  equations  as  simul- 
taneous, and  thus  use  them  for  the  determination  of /"and  n. 
In  other  words,  without  other  conditions  the  relations  of/" 
and  n  to  the  circumstances  of  the  experiment  are  indetermi- 
nate. Some  arbitrary  assumption  must  therefore  be  made  in 
regard  to  one,  and  the  other  determined  in  accordance  there- 
with. Taking  any  set  of  values  of  R  for  fixed  quality,  and 
length  at  varying  speeds,  it  seems  most  natural  to  assume 
that /will  be  constant  for  all  such  values  of  R,  and  to  then 
determine  n  in  accordance  with  this  assumption.  This  is 
equivalent  to  assuming  that  /  is  dependent  on  quality  and 
length,  and  independent  of  speed.  We  may  also  consider 
this  coefficient  as  providing  for  effects  due  to  changes  of  tem- 
perature and  density  of  liquid.  It  may  also  be  well  to  note 
that  this  coefficient,  and  in  fact  the  entire  equation,  is  con- 
sidered as  independent  of  the  depth  of  immersion. 

While  thus  taking /as  independent  of  speed  variation,  it 

s  not  independent  of  the  amount  taken  as  the  unit  of  speed. 

This  appears  as  follows:  In  the  formula,  give  v  a  value   I. 

Then  R—fA,  or  /=  R  -r-  A.     Hence  /=  the  mean  value 

of  R  per  square  foot  at  the  unit  speed. 

The  value  of /being  thus  known,  it  maybe  substituted 
in  the  various  equations  representing  the  values  of  R  at  vary- 
ing speeds,  and  the  values  of  ;/  thus  determined.  It  is  seen 
that  the  values  of  n  may  vary  with  the  speed,  and  they  will 
likewise  depend  on  the  particular  value  of /used. 

It  thus  appears  that /will  depend  on  length,  quality  of 
surface,  density,  and  temperature,  while  n  will  depend  on 
speed  and  /,  and  hence  on  speed  and  the  other  various  con- 
ditions above. 

Applying    these    principles    to    the    results    of    Froude's 


44  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

experiments  *  we  find  in  general  for  a  given  quality,  that  f 
decreases  with  increasing  length  at  a  decreasing  rate:  also 
that  f  varies  with  quality  in  the  way  we  should  naturally 
expect. 

We  turn  now  to  n  and  consider  its  relation  to  the  three 
principal  conditions,  quality,  length,  and  speed.  We  first 
note  that  we  can  form  no  scale  of  quality,  although  there  is 
plainly  a  dependence,  n  being  in  general  larger  for  rough  than 
for  smooth  surfaces.  For  length  we  find,  by  comparing  the 
results  for  constant  quality  and  speed,  that  the  relation  of  n 
to  length  is  somewhat  variable  with  the  quality.  The  results 
are  shown  graphically  in  Fig.  19.  In  (a)  and  (b)  for  varnish 
and  calico  respectively  there  is  a  falling  off,  at  first  marked 
and  then  more  gradual  as  the  length  of  50  feet  is  approached. 
In  (c)  for  fine  sand  the  value  is  practically  constant,  and  very 
nearly  so  in  (d)  for  medium  sand.  Curiously,  also,  the 
values  for  (c)  are  greater  than  those  for  (d). 

The  relation  of  n  to  speed  is  much  less  simple  or  regular. 
In  Fig.  20  are  shown  graphically  the  variations  of  n  with 
speed  for  four  qualities  as  above.  For  convenience  in  deal- 
ing with  the  data  available,  500  ft.  per  minute  was  chosen  as 
the  unit  speed.  Special  comment  on  the  curves  is  unneces- 
sary. It  may  be  stated  that  they  were  derived  by  methods 
described  in  §  78  from  the  curves  published  in  the  British 
Association  Reports  above  referred  to. 

The  use  of  these  data  for  large  high-speed  ships  involves 
the  following  considerations: 

Given  values  for /and  n  for  any  proposed  quality  of  sur- 
face, for  planes  from  2  to  50  ft.  in  length,  and  moving  at 

*  See  complete  report  in  British  Association  Reports  for  1874,  of  which 
the  table  on  page  41  is  a  partial  resume. 


RESISTANCE. 


45 


speeds  from  200  to  600  ft.  per  minute.     Required  appropriate 
values   for  ships   perhaps  600  ft.   in   length   and   moving  at 


2.1 

2.0 

a 

1.9 

Varnish 

\ 

\ 

V, 

=  ^ 



^                    

1 

— 

'• 

- 

1.8 

2.0 

&      u 

Calico 

^ 

\ 

^-^ 



— 

~ 

— 

1.0 

2.1 

c       2'° 

Fine 
Sand        i  g 



,     2.0 

d 

1  9 

Medium 

Sand 
1.8 

^^ 

,—                      — 

±^. 

J                         10                        20                        30                        40                        50 
LENGTH  IN   FEET. 

FIG.  19. 

speeds  of  20  to  25  knots  per  hour  or  2000  to  2500  ft.  per 
minute. 


46               RESISTANCE  AND   PROPULSION   OF  SHIPS 
1.9 


a 

Varnish 
1.7- 


1.8- 


2.0- 


1.9- 


b 

Calico 


1.8- 


1.7 


2.1 


2.0- 


Fine 
Sand 


1.9 


1.8- 


2.1 


d 

Medium 
Sand 


2.0- 


1.9- 


300  1400  500  600  700  800 

SPEED  IN  FEET  PER  MINUTE 

FIG.  20. 


RESISTANCE. 


47 


The  method  actually  in  use  is  either  to  assume /"and  n  as 
practically  unchanged  beyond  the  maximum  length  and  speed 
involved  in  the  experimental  data,  or  to  attempt  an  extra- 
polation to  the  given  limits,  of  the  laws  governing  their  varia- 
tion within  the  experimental  range.  The  extreme  uncertainty 
of  either  method,  especially  as  regards  the  variation  of  n,  is 
seen  by  considering  the  curves  of  Fig.  19  extended  by  judg- 
ment to  lengths  ten  times  as  great  as  the  maximum  there 
given,  or  those  of  Fig.  20  to  speeds  three  or  four  times  as 
great  as  the  highest  there  given.  The  use  of  such  methods 
of  estimation  is  only  justified  by  the  lack  of  more  extended 
information.  For  actual  computation  definite  values  must, 
of  course,  be  selected.  It  must  be  remembered,  however^ 
that  their  accuracy  for  large  ships  at  high  speeds  is  a  matter 
of  very  great  uncertainty.  Thus  in  the  tables  for  f  given 
below  it  is  doubtful  if  the  figures  in  the  third  significant 
place  have  any  real  significance  whatever.  Similarly  for  n, 
values  1.83,  1.825,  or  1.829  as  used  may  perhaps  answer 
as  well  as  any  other,  but  it  must  not  be  forgotten  that  no 
significance  can  attach  to  figures  in  the  third  place  of 
decimals,  and  but  slight  importance  to  those  in  the  second 
place. 

On  the  other  hand  we  may  remark  that  a  difference  of  I 
in  the  hundredths  of  n  will  cause  for  usual  speeds  a  difference 
of  only  about  I  per  cent  in  the  value  of  vn,  so  that  such  an 
amount  of  uncertainty  is  not  greater  than  that  to  which  we 
are  accustomed  in  most  engineering  work.  This  discussion 
shows,  however,  to  what  an  extent  extrapolation  is  necessary 
in  any  attempt  to  compute  skin-resistance,  and  how  great  is 
the  need  of  more  extended  experimental  investigation. 

In  this  connection  the  ideas  of  M.  Risbec  as  given  in  a 


48  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

recent  paper  *  will  be  of  interest.  This  author  considers  that 
it  is  more  rational  to  assume  that  fundamentally,  skin-resist- 
ance, involving  as  it  does  the  energy  of  eddying  water,  will 
be  proportional  to  the  square  of  the  speed,  and  to  write  the 
formula 

R=fAv\ 

The  coefficient  f  is  then  to  be  determined  in  accordance 
with  the  experimental  data,  and  is  considered  as  including 
a  certain  function  of  "  perturbation,"  the  value  of  which 
offsets  the  varying  values  of  n  in  the  more  usual  mode  of 
consideration.  The  possibility  of  such  a  treatment  of  the 
original  data  will  be  obvious  from  the  preceding  discussion  of 
the  relation  of  the  quantities  in  the  equation  to  the  conditions 
of  the  experiment.  We  have  simply  to  find  for  the  various 
values  of  quality,  length,  and  speed  the  quotient 


M.  Risbec  then  represents  the  quotient/  as  a  function  of 
four  parameters  for  each  quality  of  surface,  of  the  speed  v, 
and  of  the  length  L.  The  complexity  of  the  resulting  equa- 
tion and  the  uncertainty  of  the  exact  nature  of  the  param- 
eters, especially  as  affected  by  their  extension  to  lengths 
and  speeds  far  beyond  those  contained  in  the  fundamental 
data,  make  it  questionable  whether  any  advantage  is  to  be 
derived  from  this  mode  of  consideration.  It  is,  however,  of 
interest  as  showing  a  possible  mode  of  considering  the  rela- 
tion of  the  formula  to  the  characteristics  of  the  experiment. 

*  Bulletin  de  1'  Association  Technique  Maritime,  vol.  v.  p.  45. 


RESISTANCE. 


8.  SKIM-RESISTANCE  OF  SHIP-FORMED  BODIES. 


49 


While  in  the  preceding  section  we  have  referred  in  general 
to  the  application  of  the  data  therein  discussed  to  the  resist- 
ance of  ship-formed  bodies,  it  must  be  remembered  that 
fundamentally  the  data  relate  simply  to  thin  boards,  sensibly 
equivalent  to  planes. 

Now  at  any  point  of  a  ship-formed  body  let  the  horizontal 
component  of  the  tangential  force  be  denoted  by  p.  Let  ds 
be  an  element  of  area,  and  0  the  inclination  to  the  longi- 
tudinal of  the  water-lines  at  this  point.  Then  /  cos  tids  is  the 
longitudinal  component  of  the  tangential  force  or  the  tangen- 
tial resistance  for  this  element.  For  the  whole  ship  we  shall 
have,  therefore,  for  the  skin-resistance 

R-.  f  p  cos  6ds\ 

or  if p  denote  a  mean  value  of/,  we  have 
R  =  ~p  Cds  cos  0. 

The  value  of  the  integral  is  known  as  the  reduced  wetted 
surface,*  and  it  thus  appears  that  strictly  speaking  it  is  with 
this  rather  than  with  the  actual  surface  that  we  are  concerned 
in  dealing  with  skin-resistance.  The  difference  between  the 
two,  however,  is  usually  less  than  I  per  cent,  and  the  uncer- 
tainties surrounding  the  whole  question  as  already  discussed 
show  that  no  significant  error  will  be  made  by  using  either 

*  As  shown  in  the  theory  of  statical  naval  architecture  the  value  of  the 
reduced  wetted  surface  is  more  simply  expressed  by 

Reduced  surface  =f(fs  cos  Q  ^=fGdx, 

where  G  is  the  variable  girth  or  length  of  section,  and  dx  is  the  element 
of  length. 


50  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

the  one  or  the  other.  The  experiments  of  Wm.  Froude  on 
the  greyhound  led  him  to  use  for  A  the  actual  wetted  surface, 
and  the  coefficients  derived  from  his  results  and  given  in  Table 
II,  p.  53,  are  intended  to  correspond  to  this  value  of  the  area. 
In  continental  Europe  the  reduced  surface  is  frequently  used 
in  this  connection. 

The  value  of  the  wetted  surface  may  be  derived  by 
approximate  integration  as  shown  in  the  theory  of  statical 
naval  architecture,  or  by  the  use  of  one  of  the  following 
empirical  and  approximate  formulae: 

Let   S  =  wetted  surface  in  square  feet; 
D  =  displacement  in  tons- 
L  =  length  in  feet; 
H  =  draft  in  feet; 
B  =  beam  in  feet; 
b  =  block  coefficient  of  fineness. 

Then  we  may  have  . 


In  addition  to  the  above  notation  let  us  put 

k  —  coefficient  taken  from  table; 

a  «        n        tf      . 

c  —  » 

GI  =  girth  of  midship  section  ; 
/  —  prismatic  coefficient  of  fineness. 

*  Transactions  Society  of  Naval  Architects  and  Marine  Engineers,  vol. 
i.  p.  226. 

f  Transactions  Institute  of  Naval  Architects,  vol.  xxxvi.  p.  72. 


RESISTANCE. 


TABLE   I. 


Coefficient  K. 

V> 

B  +  H 

2 

2.5 

3 

4 

5 

•45 
.46 

•47 
•48 

•49 
•  50 

•  5i 
•  52 
•53 
•54 
•55 
.56 

•  57 
-58 

•59 
.60 
.61 
.62 

•  63 
.64 
.65 
.66 

.67 
.68 

.69 
.70 
•71 

.72 
•73 
•74 

•75 
•76 

•  77 
•  78 

•79 
.80 

.81 
.82 
•  83 
.84 
•  85 

1.869 
43 

1.792 
73 
57 

43 
32 
23 

15 
07 

00 

1.792 

72 

59 
46 
34 
27 

17 
ii 
02 

1.729 
21 

13 
06 
OO 

1.696 
92 

88 
84 
82 
80 
78 

76 
76 

76 
76 
76 
78 
80 
82 
84 
86 

90 

1.700 
06 
13 

19 

27 

35 
45 
56 

7i 

86 

1.806 

35 
65 

1.917 
47 
73 

1.706 
00 

1.696 

92 

88 
86 

84 
82 
80 

78 
76 
76 

76 
76 

78 
80 
82 
84 
86 
90 

94 

98 

I.  806 
1.780 
63 

49 
36 
27 
17 
ii 
05 

1.698 

94 

88 

86 
82 
80 
78 
76 
76 
76 
76 
76 
78 
82 
84 
86 
90 

1.700 

05 
ii 

17 
27 
36 
46 
61 
76 

_2L 
1.825 

50 

81 

1.696 
92 
88 
84 
80 
78 
76 
76 
76 
76 
76 
76 
78 
80 
82 
86 
90 
94 

1.699 

94 
90 
86 

84 
82 
80 
78 
78 
78 
78 
78 
82 

84 
86 
90 

94 
99. 
1.705 
ii 

17 

27 

36 

49 
61 
80 

1.702 
09 

15 
21 

29 

37 
47 
57 
70 
84 

1.700 
o? 
13 
19 
29 
39 
49 
65 
80 

1.800 

21 

49 

75 

1.904 
31 
59 
85 

1.803 
37 

1.802 

1.909 

2.OIO 

TABLE  II. 


Coefficient  c. 

B 

77 

2.O 

•933 

2.  1 

.829 

2.2 

.824 

2-3 

.820 

2.4 

.816 

2.5 

.812 

2.6 

.809 

2.7 

•  805 

2.8 

.802 

2.9 

.899 

3-0 

.896 

3-1 

•893 

3-2 

.890 

3-3 

.888 

3-4 

.885 

3-5 

.882 

3-6 

.880 

3-7 

.878 

3-8 

•875 

3-9 

•873 

4.0 

.871 

4.1 

.869 

4.2 

.867 

4-3 

•  865 

4-4 

.863 

4-5 

.860 

4.6 

•859 

4-7 

•857 

4.8 

•855 

4.9 

.853 

5-o 

•  851 

5-i 

•  850 

5-2 

.848 

5-3 

.846 

5-4 

•845 

5-5 

•843 

5-6 

.842 

5-7 

.840 

5-8 

•839 

5-9 

•837 

6.0 

.836 

6-5 

.829 

7.0 

•  823 

Intermediate  values  of  A" and  c  may  be  obtained  by  interpolation. 


52  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

Then  to  a  somewhat  greater  degree  of  accuracy  we  have 

S  = 


The  first  two  formulae  have  the  advantage  of  simplicity, 
while  the  latter  is  more  accurate,  its  errors  being  usually  less 
than  one  per  cent. 

For  the  reduced  surface  we  have,  from  the  foot-note 
on  p.  49, 

Sr  = 


With  the  same  notation  as  before  we  have  the  following 
empirical  relation  between  5  and  Sr: 


9.  ACTUAL  VALUES  OF  THE  QUANTITIES  /  AND  n  FOR 
SKIN-RESISTANCE. 

We  will  now  give  a  resum£  of  the  practical  values  which 
have  been  proposed  by  various  authorities  for  the  coefficient 
/"and  the  exponent  n. 

The  general  formula  is 

R=fAv", 

where  R  —  resistance  in  pounds; 
/=  coefficient; 

A  =.  area  of  wetted  surface  in  square  feet; 
v  =  speed  in  knots  per  hour; 
n  =  index. 

*  Transactions  Society  of  Naval  Architects  and  Marine  Engineers,  vol. 
II.  p.  297. 

}  Ibid.,  vol.  in.  p.  138. 


RESISTANCE. 


53 


TABLE   I,    SHOWING   FROUDE'S    EXPERIMENTAL   VALUES. 


Nature  of  Surface. 

Length  of  Surface. 

2  feet. 

8  feet. 

20  feet. 

50  feet. 

n 

/ 

n 

f 

n 

f 

n 

f 

2.OO 

1-95 
2.l6 

1-93 
2.OO 
2.OO 
2.OO 

.0117 
.OIIQ 
.0064 
.0281 
.0231 
.0257 
.0314 

1.85 
I.Q4 
1.99 
1.92 
2.OO 
2.OO 
2.OO 

.0121 
.0100 
.008l 
.O2O6 
.0166 
.0178 
.O2O4 

1.85 
i-93 
1.90 
1.89 

2.OO 
2.OO 
2.00 

.OIO4 

.0088 
.0089 
.0184 

•0137 
.OI52 
.0168 

1.83 

1.83 
1.87 
2.06 
2.00 

.0097 

.0095 
.0170 
.0104 
.0139 

Paraffin     

Tin-foil              

TABLE   II. 

VALUES   FOR   SHIPS   BASED   ON    FROUDE'S   EXPERIMENTS. 
(SALT  WATER.) 


Index  for  the 

Index  for  the 

Coefficient  of 

Variation  of 

Coefficient  of 

Variation  of 

Length 
in  Feet. 

Skin-resistance. 

Resistance 
with  Speed. 

Length 
in  Feet. 

Skin-resistance. 

Resistance 
with  Speed. 

f 

• 

f 

n 

8 

.01197 

1.825 

80 

.00933 

1.825 

9 

.01177 

90 

.00928 

10 

.OIl6l 

100 

.00923 

12 

.01131 

120 

.00916 

14 

.01106 

140 

.00911 

16 

.01086 

1  60 

.00907  . 

18 

.01069 

180 

.00904 

20 

.01055 

200 

.00902 

25 

.01029 

250 

.00897 

30 

.01010 

300 

.00892 

35 

.00993 

350 

.00889 

40 

.00981 

400 

.00886 

45 

.00971 

450 

.00883 

i 

50 

.00963 

500 

.00880 

< 

60 

.00950 

550 

.00877 

« 

70 

.00940 

600 

.00874 

« 

54 


RESISTANCE   AND    PROPULSION   OP   SHIPS. 


TABLE    III. 

VALUES   FOR   SHIPS   BASED    ON   TIDMAN'S   EXPERIMENTS. 
(SALT  WATER.) 


Length 
in  Feet. 

Iron  Bottom  Clean  and 
Well  Painted. 

Copper  or  Zinc  Sheathing. 

Smooth. 

Rough. 

/ 

« 

f 

n 

f 

n 

10 

.01124 

•853 

.0100 

.9175 

.01400 

.870 

20 

.01075 

.849 

.00990 

.900 

•01350 

.861 

30 

.01018 

.844 

.00903 

.865 

.01310 

•853 

40 

.00998 

•8397 

.00978 

.840 

.01275 

.847 

50 

.00991 

.8357 

.00976 

.830 

.01250 

.843 

IOO 

.00970 

.829 

.00966 

.827 

.01200 

.843 

150 

.00957 

829 

.00953 

.827 

.OJI83 

•843 

2OO 

.00944 

.829 

.00943 

.827 

.01170 

.843 

250 

.00933 

.829 

.00936 

.827 

.OII6O 

•843 

300 

.00923 

.829 

.00930 

.827 

.01152 

.843 

350 

.00916 

.829 

.00927 

.827 

.01145 

1.843 

4OO 

.OOgiO 

.829 

.00926 

.827 

.OII4O 

1.843 

450 

.00906 

,829 

.00926 

.827 

.01137 

1  843 

500 

.00904 

1.829 

.00926 

1.827 

.OII36 

1.843 

TABLE    IV. 

VALUES    FOR   PARAFFIN    MODELS    BASED    ON   TIDMAN'S 
EXPERIMENTS. 

(FRESH    WATER.) 


Index  for  the 

Index  for  the 

Coefficient  of 

Variation  of 

Coefficient  of 

Variation  of 

Length 
in  Feet. 

Skin  resistance. 

Resistance 
with  Speed. 

Length 
in  Feet. 

Skin-resistance. 

Resistance 
with  Speed. 

f 

n 

f 

n 

2 

.01176 

1.94 

12 

.00908 

i;94 

3 

.01123 

12.5 

.00901 

4 

.01083 

13 

.00895 

5 

.01050 

13  5 

.00889 

6 

.01022 

14 

.00883 

7 

.00997 

14.5 

.00878 

8 

.00973 

15 

.00873 

9 

•009S3 

16 

.00864 

10 

.00937 

17 

.00855 

10.5 

.00928 

18 

.00847 

n 

.00920 

19 

.  00840 

ii.  5 

.00914 

20 

.00834 

" 

RESISTANCE. 


55 


The  various  values  for  /  given  in  the  above  tables  are 
based  on  the  experiments  of  Froude  and  Tidman.  As  other 
special  values  we  may  mention  the  following: 

Colthurst  *  gives  the  value  /=  .0167  for  rough-sawn 
wood  and  f  •=•  .01  for  smooth-planed  wood.  The  same 
authority  states  that  for  fine  "  grass  "  or  marine  growth/ 
may  rise  to  a  value  from  .048  to  .062.  For  clean  copper  the 
value  .0073  is  given,  and  for  fresh-painted  iron  the  value  .01. 
This  last  value  is  also  that  used  by  Prof.  Rankine.  These 
values  all  relate  to  ships  of  usual  length,  or  at  least  over  50 
feet. 

In  this  connection  the  results  given  in  a  recent  paper  by 
Chief  Naval  Constructor  Hichborn  f  are  of  interest.  From 
the  data  given  in  this  paper  the  following  tabular  presentation 
is  derived:  Column  a  shows  the  percentage  increase  of  total 
resistance  with  foul  bottom  over  that  with  clean,  reckoned  on 
the  latter  as  base. 

Ship.  Speed.  a. 

Charleston 8.8  70 

Yorktown 9  72 

Philadelphia n  52 

San  Francisco 9  33 

Bennington 7  200 

Baltimore 1 1  20 

It  thus  appears,  as  we  should  expect,  that  the  term 
"  foul  "  is  entirely  indefinite,  and  may  mean  almost  any  con- 
dition in  which  the  resistance  is  sensibly  increased.  In  the 
extreme  case  of  the  table  the  total  resistance  of  the  Benning- 

*  Cited  by  Pollard  and  Dudebout,  Theorie  du  Navire,  vol.  in.  p.  376. 
•f-  Transactions  Society  of  Naval  Architects  and  Marine  Engineers,  vol. 
n.  p.  159- 


56  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

ton  when  foul  is  three  times  that  when  clean.  At  the  low 
speeds  at  which  these  runs  were  made,  most  of  the  resistance 
will  be  due  to  the  skin,  so  that  while  the  above  figures  refer 
to  total  resistance,  the  ratios  for  the  skin-resistance  will  not 
be  far  different,  though  slightly  greater. 

These  possibilities  impress  in  the  strongest  manner  the 
extreme  importance  of  a  clean  bottom  especially  for  economi- 
cal cruising  at  moderate  speeds,  and  for  the  possible  attain- 
ment of  the  highest  speeds. 

10.  WAVES. 

As  introductory  to  the  subject  of  wave-resistance  we  shall 
give  in  the  present  section  a  brief  account  of  the  commonly 
accepted  theories  of  wave-motion  in  liquids,  with  a  statement 
of  the  more  important  results,  but  without  the  details  of  the 
mathematical  development.* 

In  order  to  examine  in  a  satisfactory  manner  the  influence 
of  waves  upon  a  ship,  some  working  theory  as  to  their 
dynamical  constitution  is  required.  It  must  be  understood 
that  such  a  theory  is  not  to  be  viewed  as  final  truth  in  itself, 
but  rather  as  a  convenient  method  of  correlating  as  nearly  as 
possible  the  known  facts,  and  of  deducing  by  appropriate 


*  Among  others,  the  following  works  may  be  consulted  by  those  in- 
terested: 

Rankine  et  al.:  "Shipbuilding,  Theoretical  and  Practical."  London, 
1866. 

J.  Scott  Russell:  "  System  of  Modern  Naval  Architecture,"  vol.  i.  Lon- 
don, 1865. 

Encyclopedia  Britannica,  article  Wave. 

Pollard  et  Dudebout:  Theorie  du  Navire,  vol.  in.     Paris,  1892. 

Gatewood  :  Journal  U.  S.  Naval  Institute  1883,  p.  223.     Annapolis. 

Stokes:  Cambridge  Transactions  1847. 

Encyclopedia  Metropolitana,  article  Tides  and  Waves. 


RESISTANCE. 


57 


mathematical  means  the  probable  nature  and  extent  of  the 
influences  which  are  to  be  examined.  The  most  simple  of  the 
theories  proposed  to  this  end  is  that  known  as  the  trochoidal. 
The  theory  receives  its  name  from  the  trochoid,  which  is  the 
form  of  the  contour  of  waves  resulting  from  this  mode  of 
viewing  their  constitution.  The  trochoid  may  be  defined  as 
the  locus  of  a  point  P,  Fig,  21,  which  revolves  uniformly 


21. 


about  a  center  O,  which  center  itself  moves  uniformly  along 
a  straight  line  OOa.  In  this  case  if  O  moves  uniformly 
along  OOf,  and  OP  revolves  uniformly  counter-clockwise, 
the  curve  P9Pt  .  .  .  P6  will  result.  For  comparison  there  is 
shown  in  dotted  lines  a  sinusoid  of  equal  length  and  height, 
and  the  difference  between  the  two  is  indicated  by  the  shaded 
area. 

A  trochoid  may  also  be  defined  as  the  locus  of  a  point  in 
the  radius  of  a  circle,  which  circle  rolls  uniformly  along  a 
straight  line.  Thus  in  Fig.  22  let  a  circle  of  radius  OQ  roll 
along  on  the  under  side  of  AB.  Let  P  be  the  tracing-point. 
Then  the  locus  of  /'will  be  a  curve  CPDE,  and  the  identity 
in  character  between  the  two  curves  in  Figs.  21  and  22  is 
readily  seen.  If  the  tracing-point  Pis  on  the  circumference 
of  the  circle  the  locus  becomes  the  common  cycloid  as  a 
special  case  of  the  trochoid.  If  the  point  passes  beyond  Q, 
as  to  R,  the  locus  is  still  called  a  trochoid.  The  specific 


5*>  RESISTANCE  AND   PROPULSION   OF  SHIPS. 

names  prolate-trochoid  and  curtate-trochoid  are  given  to 
these  two  classes  of  curves.  It  is  only  with  the  former  as 
shown  in  the  figures  that  we  are  here  concerned. 

Waves  whose  contours  are  very  nearly  trochoidal  are  but 
rarely  met  with,  though  a  moderately  heavy  swell  in  calm 
weather  at  a  considerable  time  after  the  subsidence  of  the 
storm  to  which  it  owes  its  existence,  quite  closely  conforms 


\ 


FIG.  22. 


to  this  type.  The  trochoidal  wave  is,  moreover,  the  type- 
most  readily  examined  by  mathematical  means,  and  is  there- 
fore naturally  taken  for  purposes  of  theoretical  investigation. 
The  ideal  constitution  of  a  trochoidal  wave  involves  the 
following  suppositions: 

(1)  That  for  any  given  wave  all  particles  revolve  in  fixed 
circular  orbits,  in  vertical  planes  parallel  to  the  direction  of 
propagation  and   normal   to   the  lines  of  crests  and  hollows, 
with  the  same  constant  angular  velocity,  and  in  direction  with 
the  propagation  when  in  a  crest. 

(2)  That   for  any  layer  of  water  originally  horizontal   the 
radii   are   equal  and  the  centers  are  on  the  same  horizontal, 
line. 


RESISTANCE.  59 

(3)  That  the  radii  decrease  with  increase  of  depth  accord- 
ing to  a  law  to  be  stated  later.    ' 

(4)  That  all  particles  originally  in  the  same  vertical  are 
always  in  the  same  phase. 

It  follows  that  the  surface  layer  of  particles  and  all  suc- 
cessive layers  originally  horizontal  will  take  the  form  of 
trochoidal  surfaces  of  continually  decreasing  altitudes  accord- 
ing to  the  law  of  decreasing  radii.  It  further  results  that  all 
corresponding  crests  and  hollows  for  successive  layers  will  lie 
in  the  same  vertical  line,  and  hence  that  all  wave-lengths  for 
the  successive  trochoidal  subsurfaces  will  be  equal,  and  that 
the  velocity  of  propagation  for  all  must  be  the  same. 

This  ideal  constitution  of  a  wave  may  be  illustrated  in 
Fig.  23.  The  diagram  represents  a  section  of  a  certain  por- 


h  I 

FIG.  23. 

tion  of  still  water  divided  by  horizontal  and  vertical  planes 
into  rectangular  blocks  as  shown  in  dotted  lines.  The  full 
lines  represent  the  same  blocks  of  water  when  thrown  into 
wave-motion.  The  horizontal  planes  o,  1,2,  etc.,  for  still 
water  are,  in  the  wave,  thrown  into  the  trochoidal  surfaces 
a,  b,  r,  etc.  The  corresponding  circular  orbits  are  shown  on 


60  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

the  left,  and  their  centers  are  seen  to  be  elevated  above  the 
corresponding  still-water  planes  by  successively  decreasing 
amounts  as  we  go  from  the  surface  downward.  To  avoid 
confusion  in  the  diagram  the  circular  orbits  are  elsewhere 
omitted.  The  motion  involved  in  the  figure  may  be  viewed 
from  several  standpoints.  Thus  we  may  consider  the 
trochoidal  layers  and  note,  as  the  wave  progresses,  the  change 
in  thickness  and  form  at  any  one  point.  Or  again  we  may 
consider  the  vertical  columns  of  water  between  .the  distorted 
verticals  af,gh,  £/,  etc.,  and  note  that  as  the  wave  progresses 
these  columns  shorten  and  broaden  and  then  lengthen  and 
narrow,  at  the  same  time  waving  to  and  fro  with  the  period 
of  the  wave. 

Again,  each  rectangular  block  of  still  water  is  represented 
in  the  wave  by  a  distorted  block,  and  the  successive  forms 
into  which  one  of  these  blocks  is  thrown  may  be  seen  by  fol- 
lowing along  the  series  of  distorted  quadrilaterals  between  the 
pair  of  corresponding  trochoidal  surfaces  for  the  wave. 

In  order  that  such  a  wave  may  actually  exist  in  a  liquid  it 
must  fulfil  certain  hydrodynamic  conditions.  These  are: 
(i)  That  at  the  upper  surface  the  pressure  must  be  constant 
and  normal  to  the  surface.  (2)  That  the  liquid  cannot 
undergo  expansion  or  compression  during  the  process  of 
wave-propagation.  These  conditions  may  be  made  to  furnish 
a  relation  btween  the  length  and  the  velocity  of  propagation, 
and  a  law  for  the  decreasing  radii  of  the  circular  orbits. 
These  are  as  follows: 

Let  L  =  length  in  feet; 

V  =.  linear  velocity  in  feet  per  second; 
R  —  radius  of  rolling  circle: 
GO  =  angular  velocity; 


RESISTANCE. 


61 


T=  periodic  time; 

g  =  value  of  gravity  =  32.2, 


Then  V -- 

or  V  —  2.265  VL  ft.  per  sec., 

and  F—  1.341  VL  knots  per  hour, 

also  T- 


.  .  (i) 


(2) 


Let  ra  =  radius  of  orbit  at  surface; 

r  =  radius  of  orbit  at  depth  of  center  z  below  center 

of  surface  orbits; 
z  •=.  any  given  depth  of  orbit  center  below  center  of 

surface  orbits; 
e  =  base  of  Naperian  logarithms. 


Then 


or 


R 

Z  2W3 

r  —  rQe   &  =  r0e     L 


(3) 


If  these  various  conditions  are  fulfilled,  therefore,  it 
appears  that  the  geometrical  motion  here  supposed,  if  once 
inaugurated,  is  hydrodynamically  possible.  This  does  not 
prove  that  in  any  actual  wave  the  particles  move  as  herein 
assumed.  Observation  shows,  however,  that  these  supposi- 
tions are  near  the  truth,  and  the  actual  motion  and  character- 
istics of  waves,  so  far  as  admitting  measurement,  are  for  the 
most  part  in  fairly  satisfactory  agreement  with  those  resulting 
from  the  type  of  wave-motion  assumed.  It  seems,  therefore, 
reasonable  to  accept  the  theory  as  a  working  basis  for  the 


62  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

investigation  of  such  characteristics  of  wave-motion  as  we  are 
here  concerned  with. 


We  may  now  state  a  further  series  of  results  following 
upon  the  conditions  assumed. 

The  point  of  maximum  slope  of  any  trochoid  corresponds 
to  a  value  of  3  given  by 

cos  0  =     , (4) 


or 


R 

If  0  is  this  maximum  slope,  then 

r 


tan  0  = 


4/A2  -  ra' 

and         sin  0  =  -=•  =  — , (5) 

where  h  =  height  of  wave  =  2r. 


The  normal  at  any  point  intersects  the  vertical  through 
the  center  of  the  rolling  circle  at  the  constant  distance  R 
above  it,  and  therefore  always  at  the  point  of  contact  of  the 
rolling  circle  with  the  line  on  which  it  rolls. 


The  pressure  on  any  trochoidal  subsurface  is  constant  and 
equal  to  that  due  to  the  depth  of  the  corresponding  plane 
surface  in  still  water. 

Let  z  =  depth  of  the  line  of  centers; 
rQ  —  radius  of  surface  orbit; 
r  —  '  orbit  with  center  at  depth  z-, 

R=      "       "  rolling  circle. 


KESIS  TA  NCE. 


Then  the  pressure  in  the  crest  of  a  wave  is  less  than  that 
for  an  equal  depth  in  still  water  in  the  ratio 


2  — 


2R 


—  r 


(6) 


Similarly  the  pressure  in  the  hollow  of  a  wave  is  greater 
than  that  for  an  equal  depth  in  still  water  in  the  ratio 


(7) 


At  any  point  the  resultant  force  due  to  gravity  and  to 
centrifugal  force  acts  along  the  normal  and 
hence  through  the  instantaneous  center  of 
motion  of  the  rolling  circle  as  specified 
above.  In  Fig.  24  let  OA  =  R.  Then  if 
the  force  due  to  gravity  is  represented  by 
OA  or  R,  that  due  to  centrifugal  force  will 
be  represented  by  the  radius  OP,  and  the 
resultant  by  the  line  AP. 


Let  ;//  =  mass  of  particle; 

g  =  value  of  gravity  =  32.2  ; 


FIG.  24. 


GO  =  angular  velocity  —  \  /  — -  ; 

/=  resultant  force  represented  by  AP. 
Then  we  have 


/  =  m  !£•' 


'  -  2gro?  cos  0.       .      .      .     (8) 


'he  line  of  orbit  centers  for  any  trochoidal  surface  or 


64 


RESISTANCE   AND    PROPULSION  OF  SHIPS. 


sheet    is   raised    above   the   original   still-water    level    of   the 
corresponding  plane  by  the  distance 


71 '  T  T 

T  =  ~2~R 


2R 


(9) 


Let  H  denote  the  still-water  depth  of  a  layer  correspond- 
ing to  the  trochoidal  sheet  with  line  of  centers  at  a  depth  z 
below  the  center  line  of  surface  orbits.  Then 


2R 


(10) 


The  total  energy  of  a  wave  is  always  half  kinetic  and  half 
potential. 

Let  E  denote  total  energy  in  foot-pounds; 
cr       "       density; 
L       "       length  in  feet; 
r0       "       radius  of  orbit  at  surface; 
h       li       height  of  wave  =  2r0; 
R       "       radius  of  rolling  circle. 

_       «•/>.' 


Then 


or         E  = 


vU? 

8 


l  (k\'\ 

(i-  4-935  (j)). 


If  we  put  <r  =  64,  we  have 

E  =SLtfi-  4-935 


The  total  power  involved  in  a  series  of  waves  per  foot  of 
breadth  is 

Power  =  .0329  VLP  (i  -  4-935  (j)  )•    •     •     (12) 


RESISTANCE.  05 

Combinations  of  Wave  Systems. — Given  two  trochoidal 
systems  in  general  with  parallel  crests,  moving  in  the  same 
direction.  The  result  of  such  a  combination  geometrically, 
as  is  well  known,  is  to  produce  a  series  of  groups  of  waves, 
each  group  reaching  a  maximum  of  height  where  two  crests 
correspond,  and  a  minimum  where  crest  and  hollow  combine. 
Each  group  therefore  culminates  in  a  wave  of  maximum  alti- 
tude with  comparatively  small  disturbance  betweeen  the  suc- 
cessive groups.  It  may  be  shown  that  such  geometrical 
combination  does  not  fulfil  the  necessary  hydrodynamical 
conditions,  and  therefore  cannot  exist  permanently  as  a  com- 
bination of  actual  trochoidal  systems.  The  actual  result  of  the 
combination  of  two  wave  systems  is  to  produce  a  more  or  less 
mixed  resultant  system,  with  characteristics  relative  to  which 
those  given  by  the  geometrical  combination  of  trochoidal  sys- 
tems are  intended  simply  as  approximations.  We  do  find, 
however,  actual  combinations  of  wave  systems  agreeing  in  all 
their  general  characteristics  with  those  given  by  the  geometri- 
cal combination  referred  to,  and  it  therefore  seems  fair  to  use 
the  method  as  a  working  basis  for  the*  examination  of  such 
points  as  we  are  interested  in. 

As  a  special  case  let  us  assume  that  the  two  series  are  of 

Components 

£ 

Resultant 

FIG.  25. 

equal  altitude,  //,  but  of  different  lengths,  Z,  and  Za.  Then 
the  geometrical  combination  will  give  a  result  as  indicated  in 
Fig.  25.  At  A  and  C  the  phases  are  opposite,  and  the  values 


66  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

of  h  being  equal  the  resultant  disturbance  is  o,  and  we  have 
for  a  little  space  practically  undisturbed  water.  At  a  point 
B  midway  between,  the  phases  are  the  same  and  the  two 
effects  are  added,  giving  a  wave  of  height  2/z,  the  maximum 
possible  in  the  entire  series.  The  group  from  A  to  C  is  then 
indefinitely  repeated  to  the  right  and  left  as  far  as  the  given 
conditions  hold. 

For  such  a  group  the  length  AC  is  given  by  the  equation 


Let  v  =  velocity  of  propagation  of  the  resultant  group  as 

a  whole; 
vl  =  velocity   of    propagation   of  the    component   of 

length  Z,  ; 
#a  =  velocity   of   propagation  of   the   component   of 

length  L^. 


TM 

Then 


',  +  *,' 


I      II 

or  -  =  — —  . 

rrt  nt  l         *Jt 


As  L^  approaches  L^  in  value  the  length  of  the  group  A  C 
will  increase  and  the  velocity  will  approach  more  and  more 
nearly  to  one  half  that  of  the  components.  At  the  limit 
where  Ll  =  L^  the  length  would  become  indefinite  and  the 
velocity  would  become  one  half  that  of  the  components. 
Since  any  given  group  is  the  expression  of  a  certain  distribu- 
tion of  energy,  it  is  considered  that  the  velocity  of  propaga- 
tion of  the  energy  is  the  same  as  that  of  the  group. 


RESISTANCE. 


67 


We  will  now  take  the  special  case  of  two  trochoidal 
systems  of  different  altitudes,  but  of  equal  lengths  and  there- 
fore of  equal  velocities. 

In  Fig.  26  let  O  denote  the  orbit  center  for  a  given  sur- 
face particle  as  affected  by  either  component.  Strictly 
speaking  the  two  centers  are  not 
at  the  same  level,  but  the  differ- 
ence is  very  small  and  is  usually 
neglected.  At  any  given  instant 
of  time  let  DO  A  denote  the  phase- 
angle  6l  of  the  first  component,  and 
DOB  the  phase-angle  ^  of  the 
second  component.  Then  AOB 
will  represent  the  constant  differ- 
ence of  phase  between  the  two 
components.  Also  let  OA  =  rt  and 
OB  =  r9.  Then  the  combination 
of  the  two  components  will  result  in  a  trochoidal  system  of 

wave-length  equal  to  that  of  its  components,  and  of  surface 
orbit  radius  r  =  OC,  and  with  phase  relationships  with  its 
components  as  given  by  the  angles  CO  A  and  COB.  In  other 
words,  if  the  parallelogram  OACB  is  made  to  revolve  uni- 
formly about  O  while  the  latter  moves  uniformly  along  x, 
the  two  speeds  being  appropriate  to  the  wave-length  taken, 
then  the  point  A  will  trace  the  contour  of  the  first  com- 
ponent, B  that  of  the  second,  and  C  that  of  the  resultant. 

Let  a  denote  the  linear  difference  between  similar  points 
on  the  two  components,  GO  the  angular  velocity,  V  the  linear 
velocity,  and  R  the  radius  of  the  rolling  circle.  Then  the 
angle  denoting  the  phase  difference  AOB  is  given  by 


FIG.  26. 


68  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

(&a        a        27ta 
AOB  =  —  =  -=•  =  -7—,  in  angular  measure, 

or  AOB  —  — j —  in  degrees. 

From  the  triangle  OAC  we  derive  for  the  altitude  of  the 
resultant  system 


*.  cos  .    •      •      .      (15) 


As  to  the  degree  of  fulfilment  which  the  ideal  theory- 
rinds  in  actual  waves  it  may  be  said  in  general,  and  without 
entering  into  details,  that,  so  far  as  observations  can  be  made, 
most  of  the  actual  characteristics  are  in  fair  agreement  with 
the  results  derived  from  theory.  The  agreement  is  by  no 
means  complete,  and  in  fact  differs  in  certain  particulars  in 
such  way  as  to  lead  to  the  belief  that  a  truly  trochoidal  wave 
is  but  rarely  if  ever  met  with.  On  the  other  hand  the  agree- 
ment is  sufficiently  close  to  seemingly  warrant  the  use  of  the 
theory  for  all  purposes  of  the  naval  architect,  at  least  in  so 
far  as  he  is  concerned  with  waves  in  deep  water. 


In  regard  to  the  limiting  sizes  of  waves  the  following 
results  may  be  given: 

From  reliable  observation  it  may  be  fairly  concluded  that 
waves  longer  than  1200  to  1500  feet  are  very  rarely  met  with, 
though  there  seems  ground  for  accepting  reports  of  excep- 
tional waves  of  a  length  approaching  3000  feet.  The  corre- 
sponding periods  would  be  15  to  17  seconds  for  the  first 
named,  and  about  24  seconds  for  the  last. 

With  regard  to  height  the  evidence  is  much  more  conflict- 
ing. It  appears,  however,  that  from  40  to  45  feet  is  about 


RESISTANCE. 


the  maximum  value  which  can  be  accepted  as  reliable,  and 
values  approaching  these  figures  are  met  with  but  rarely. 
From  25  to  30  feet  may  be  taken  as  the  ordinary  limit.  It 
should  be  noted,  however,  that  these  heights  relate  to  a 
series  of  regular  waves  as  contemplated  by  the  theory,  and 
not  to  exceptional  results  arising  from  the  interference  and 
combination  of  different  systems. 

With  regard  to  ratio  of  height  to  length,  it  is  found 
usually  to  decrease  with  increase  of  length.  The  highest 
values  of  this  ratio  appear  to  be  about  I  :  6,  such  being  found 
only  with  comparatively  short  waves.  More  commonly  the 
ratio  is  from  I  :  12  to  I  :  25.  The  corresponding  values  of 
the  maximum  inclination  are  respectively  about  32°,  15°, 
and  7°. 

Shallow-water  Waves. — When  we  come  to  waves  in 
shallow  water  we  find,  as  nearly  as  can  be  determined,  the 
paths  of  the  particles  to  be  elliptical  with  the  longer  axis 
horizontal.  In  order  to  meet  this  departure  from  the  tro- 
choidal  system,  the  so-called  ellip- 
tical trochoidal  or  shallow-water 
wave  system  is  used.  This  ideal 
system  may  be  generated  as  fol- 
lows: Given  an  ellipse,  Fig.  27, 
with  its  major  axis  horizontal, 
moving  uniformly  along  X  without 
angular  change.  Draw  any  line 
OB  at  an  angle  6  with  the  verti-  FlG-  27- 

cal.  Through  B  draw  a  vertical  and  through  A  a  hori- 
zontal. They  will  intersect  on  the  ellipse  at  P.  Let  OB 
revolve  uniformly  to  the  left.  Then  the  locus  of  the  point  P 


7<D  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

thus  determined  by  the  movement  of  the  ellipse  along  OX 
and  of  OB  about  O  will  be  an  elliptical  trochoid.  Comparing 
this  contour  with  that  resulting  from  the  circle  as  a  base,  as 
in  Fig.  28,  the  former  is  seen  to  lie  slightly  within  the  latter, 
thus  making  a  flatter  hollow  and  steeper  crest,  these  differ- 
ences becoming  more  pronounced  as  the  difference  between 
the  two  axes  of  the  ellipse  is  greater 


Full  line,    Circular  Trochoid 
Dotted  line,    Elliptical  Trochoid 

FlG,    28. 

In  the  present  connection  the  feature  of  especial  interest 
is  the  velocity  of  propagation.  For  this  the  following  ex- 
pression may  be  derived: 

Let  V •=•  velocity  in  feet  per  second; 
r0  =  semi-major  axis  at  surface; 
«0=       "    minor     "      "     " 
L  =  wave  length ; 
g  =  value  of  gravity  =  32.2. 


Then 


r= 


(16) 


This  equation  shows  by  comparison  with  (i),  that  the 
velocity  of  a  shallow- water  wave  is  less  than  that  of  a,  deep- 
water  wave  of  equal  length,  a  conclusion  borne  out  by 
experience. 

Let  d  =  depth  of  the  water.  Then  when  2nd  ~-  L  is 
small  it  may  be  shown  that 


v= 


(-7) 


RESISTANCE. 


It  should  be  noted  that  the  ideal  wave  system  thus 
derived,  does  not,  and  seemingly  cannot,  be  made  to  fulfil  the 
hydrodynamic  conditions  necessary  to  its  actual  existence  as 
assumed.  This  does  not  prevent,  however,  such  form  of 
motion  from  being,  possibly,  a  close  approximation  to  what 
actually  takes  place,  nor  does  it  seriously  interfere  with  its 
usefulness  as  an  approximate  working  theory  of  shallow-water 
waves. 

Waves  of  Translation. — The  waves  thus  far  considered 
occur  naturally  in  indefinite  series,  and  represent  the  result 
of  a  widely  distributed  disturbance.  The  typical  wave  of 
translation  is  individual  in  character,  and  represents  the  result 
of  a  local  disturbance.  In  Fig.  29  let  AB  be  a  trough  or  canal 

cc,          


D  D, 

FIG.  2g. 

of  a  length  AB,  large  relative  to  its  breadth  and  depth.  Let 
CD  be  a  false  end  or  partition  movable  within  the  canal 
along  the  direction  AB.  If  now  CD  be  moved  rapidly  from 
CD  to  C,Dlt  the  water  will  become  at  first  banked  up  in  front 
of  it;  but  as  CD  is  retarded  and  finally  brought  to  rest  at 
£*,/?,.  the  banked-up  water  will  leave  C]Dl  and  travel  on  as  a 
crest  or  hump  elevated  entirely  above  the  general  level  of  the 
tank,  somewhat  as  indicated  in  the  diagram.  Such  a  wave  is 
called  positive. 

Let  us  again  suppose  that  the  partition  was  at  ClDl  and 
the  water  at  rest.      Let  it  then  be  moved  from  C,D,  to  CD. 


72  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

As  a  result  a  hollow  or  depression  will  be  formed  existing 
wholly  below  the  general  level  of  the  surface.  This  hollow 
will  then  be  propagated  on  in  a  manner  similar  to  the  crest 
previously  described.  Such  a  wave  is  called  negative. 

A  positive  wave  may  also  be  formed  by  plunging  into  the 
water  at  one  end  of  the  canal  a  block  which  will  occupy  the 
volume  CDDfv  or  by  the  sudden  introduction  of  a  certain 
additional  amount  of  liquid  into  the  end  of  the  trough. 
Similarly  a  negative  wave  may  be  formed  by  withdrawing  a 
block,  or  by  withdrawing,  as  by  a  lifting-pump,  a  certain 
amount  of  water. 

Positive  waves  if  properly  formed  are  quite  permanent, 
only  slowly  losing  their  energy  and  form  by  external  friction 
and  internal  viscous  forces.  If  not  properly  formed  the  wave 
will  soon  break  up  into  an  irregular  series,  the  members  of 
which  are  propagated  with  different  velocities,  so  that  the 
energy  is  soon  dissipated  and  the  form  disappears.  The 
negative  wave  is  much  less  permanent  than  the  positive,  and 
can  be  made  to  preserve  its  integrity  for  a  short  time  only. 

The  horizontal  movement  of  any  sheet  of  particles  sit- 
uated in  a  transverse  plane  seems  to  be  the  same.  The 
vertical  motion  varies  according  to  its  distance  from  the 
bottom  of  the  canal,  and  seems  to  be  very  nearly  proportional 
to  such  distance.  The  path  of  any  particle  seems  to  be  very 
nearly  a  semi-ellipse,  as  shown  at  AGO  in  Fig.  30.  It  there- 
fore appears  that,  due  to  the  passage  of  the  wave,  a  particle 
which  was  formerly  at  A  has  been  transported  and  left  at  O; 
furthermore,  that  all  particles  in  the  transverse  plane  of  A  will 
undergo  the  same  longitudinal  translation,  and  will  be  left  in 
a  transverse  plane  containing  O.  Care  must  be  here  taken 
to  distinguish  between  the  motion  of  the  particle  and  the 


RESISTANCE, 


73 


motion  of  propagation.  While  the  particle  moves  through 
its  path  as  AGO,  the  wave-form  will  move  forward  its  wave- 
length OF  or  LM,  Fig.  29. 


FIG.  30. 

The  approximate  contour  of  a  wave  of  translation  may  be 
determined  by  the  following  construction: 

Let  AGO  be  the  ellipse  representing  the  path  of  a  surface 
particle,  and  let  AH  be  the  depth  of  the  canal.  Then  lay  off 
AF=  27th  and  construct  the  sinusoid  AEF.  Next  lay  off  to 
the  right  from  points  on  this  curve,  as  A,  c,  g,  etc.,  distances 
AO,  ab,  ef,  etc.,  or  the  intercepts  between  the  ellipse  and 
the  vertical  OP.  The  points  Oy  d,  z,  etc.,  thus  determined 
will  give  the  desired  contour  of  the  ideal  wave  of  translation 
corresponding  to  the  conditions  assumed. 


Let  v  =  velocity  of  propagation ; 

//  =  depth  of  water  when  at  rest ; 

k  =  height  of  wave  above  still-water  level ; 

L  =  wave-length  =  OF,  Fig.  30 ; 

a  —  range  of  translation  =  AO.  Fig.  30; 

V  —  volume  of  water  constituting  wave: 


74  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

b  =  breadth  of  canal; 
A  =  area  of  profile  of  wave. 


Then          v  =  Vg(h  +  k) (18) 

When  k  is  relatively  small  we  have 

v—^/g/i (19) 

It  is  found  that  k  cannot  exceed  k.  On  reaching  this 
value,  or  before,  the  wave  breaks  at  the  crest.  Hence  as  a 
limiting  value  of  v  we  have 

v  —  V~2gh. .     (20) 

For  the  wave-length 

L  —  27th  —  a, (21) 

or  where  OL  is  relatively  small 

L  =  2nh (22) 

For  the  value  of  A  we  have 

A  =  ofh (23) 

As  with  oscillatory  waves,  the  energy  is  one-half  kinetic 
and  one-half  potential,  the  entire  amount  being 

Energy  =  2<rVz\ (24) 

where  0"  =  density; 
V=  volume; 

J  =  elevation  of  center  of  gravity  of  wave  above  the 
still-water  level. 


RESISTANCE. 


75 


TABLE  I. 

SHOWING   RELATION   BETWEEN   Z,    T,   AND    V  FOR 
TROCHOIDAL  WAVES. 


L 

T 

Fin  Feet 
per  Sec. 

Fin  Knots.1]        L 

T 

Fin  Feet 
per  Sec. 

Fin  Knots. 

IO 

1.40 

7.2 

4.2 

350 

8.30 

42.4 

25-1 

2O 

2.00 

10.  1 

6.0 

4OO 

8.80 

45-3 

26.8 

30 

2.42 

12.4 

7-3 

450 

9.40 

48.0 

28.4 

40 

2.80 

14-3 

8-5 

500 

9.90 

50.6 

30.0 

50 

3.10 

16.0 

9-5 

550 

10.40 

53-1 

31-4 

60 

3-40 

17.6 

10.4 

6OO 

10.80 

55-5 

32.8 

70 

3-70 

19.0 

II.  2 

650 

ii.  20 

57-8 

34-2 

80 

4.00 

20.2 

I2.O          |        7OO 

11.70 

59-9 

35-5 

90 

4.20 

21.5 

12.7 

750 

12.  IO 

62.0 

36.7 

100 

4.40 

22.7 

13-4 

800 

12.50 

64.1 

37-9 

150 

5-40 

27-7 

I6.4 

850 

I2.9O 

66.0 

39-1 

200 

6.20 

32.0 

ig.O 

9OO 

I3.2O 

68.0 

40.2 

250 

7.00 

35-8 

21.2 

950 

13.60 

69.8 

41-3 

300 

7.60 

39-2 

23-2 

1000 

14.00 

71.6 

42.4 

TABLE  II. 


SHOWING  VALUES  OF  —  FOR  VARYING  VALUES  OF 


z 

f 

z 

r 

z 

r 

z 

r 

L 

7*0 

L 

ro 

L 

*-o 

L 

r0 

.01 

•9391 

.07 

.6442 

.25 

.2079 

.60 

.0231 

.02 

.8820 

.08 

.6049 

•30 

.1518 

.70 

.0123 

•03 

.8283 

.09 

.5681 

•35 

.1109 

.80 

.0066 

.04 

•7778 

.10 

.5336 

.40 

.0810 

.90 

.0035 

•05 

•7304 

•15 

.3897 

•45 

.0592 

1.  00 

.0019 

.06 

.6859 

.20 

.2846 

•  50 

.0432 

76 


RESISTANCE  AND   PROPULSION  OF  SHIPS. 


TABLE  III. 

SHOWING  THE  VARIATION   OF   PRESSURE  VERTICALLY  DOWN- 
WARD  IN   THE  CREST  AND   HOLLOW   OF   A  WAVE. 


z 
~L 

Crest. 

Hollow. 

Actual  Depth 

Value  of  Ratio 

LIO]  (6). 

Actual  Depth 

Value  of  Ratio 

[ioj  (7)- 

L 

L 

•  05 

•0635 

•730 

.0365 

.269 

.10 

.1233 

.766 

.0767 

.231 

•  15 

.1805 

-794 

.0836 

.200 

.20 

.2358 

.8l8 

.1642 

.174 

.25 

.2896 

.837 

.2105 

.152 

•30 

.3424 

.854 

.2576 

•135 

•35 

•3945 

.868 

•3055 

.I2O 

.40 

.4460 

.879 

•3540 

.108 

•45 

•4975 

.889 

.4025 

.098 

•  50 

•  5478 

.898 

.4521 

.088 

.60 

.6488 

•9!3 

•5512 

•075 

.70 

•7493 

.924 

.6507 

.064 

.80 

.8497 

•932 

.7502 

.056 

.90 

.9498 

•939 

.8502 

.049 

1.  00 

1.0499 

.946 

.9501 

•045 

1.50 

1.6000 

•963 

1.4500 

.029 

2.0O 

2.0500 

.972 

1.9500 

.022 

TABLE  IV. 

SHOWING   POWER   OF   OCEAN  WAVES   IN  HORSE-POWER  UNITS. 


Length  of  Wave  in  Feet. 


k 

25 

50 

75 

100 

150 

200 

300 

400 

50 

.04 

•23 

.64 

1.31 

3.62 

7-43 

20.46 

42 

40 

.06 

.36 

1.  00 

2.05 

5.65 

H.59 

31.95 

66 

30 

.12 

.64 

1.77 

3-64 

10.02 

20.57 

56.70 

116 

20 

.25 

1.44 

3.96 

8.13 

21-79 

45.98 

126.70 

260 

15 

.42 

2.83 

6.97 

14.13 

39-43 

80.94 

223.06 

457 

10 

.98 

5-53 

15.24 

31.29 

86.22 

177.00 

487.75 

1001 

RESISTANCE. 


77 


TABLE  V. 

COMPARISON  BETWEEN  WAVES  IN  SHALLOW  AND  DEEP  WATER. 


Depth  of  Water  as 
a  Fraction  oi  the 
Wave-length. 

Ratio  between  Quantities  for  Shallow  Water  and  Corresponding 
Quantities  ior  Deep  Water. 

Ratio  of  Axes  of 
Surface  Dibits. 

Length  for  Given 
Velocity. 

Velocity  for  Given 
Length. 

.OI 

.063 

15.90 

.251 

.02 

.124 

8.08 

•352 

•03 

.186 

5.376 

•431 

.04 

.246 

4-065 

.496 

•05 

.304 

3-289 

•552 

•075 

•439 

2.2/7 

.663 

.10 

•  557 

.796 

.746 

.15 

.736 

•358 

.858 

.20 

•  847 

.180 

.920 

•25 

.917 

.091 

•958 

•30 

•955 

047 

•  977 

•35 

•  975 

.026 

.987 

.40 

.987 

.013 

•993 

•45 

•993 

.007 

•996 

•50 

.996 

.004 

.998 

.60 

•999 

.OOI 

•999 

•75 

•9999 

.OOOI 

•9999 

1.  00 

•99999 

.OOOOI 

•999999 

11.  WAVE-FORMATION  DUE  TO  THE  MOTION  OF  A  SHIP- 
FORMED  BODY   THROUGH    THE   WATER. 

In  §  2  it  is  shown  that  if  the  surface  of  the  water  were 
covered  by  a  rigid  plane  through  which  the  ship  could  pass 
without  resistance,  and  which  closed  after  it,  no  waves  could 
be  formed,  but  a  disturbance  in  the  distribution  of  pressure 
would  result,  such  that  about  the  bow  and  stern  there  would 
be  an  excess  and  about  the  middle  portions  a  defect.  If  the 
hypothetical  rigid  plane  is  removed,  we  shall  have  instead  of 
such  a  modification  of  pressure,  and  as  the  natural  manifes- 
tation of  the  tendency  to  produce  it,  an  elevation  of  the 
surface  about  the  bow  and  stern  and  a  depression  about  the 


7°  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

middle  portions.  The  result  of  these  initial  or  fundamental 
tendencies  is,  however,  much  modified  by  propagation  and 
the  interference  or  combination  of  their  elementary  results. 


In  any  event,  however,  the  distribution  of  normal  pressure 
over  the  surface  of  the  ship  will  be  much  changed  from  that 
corresponding  to  hydrostatic  conditions,  such  change  result- 
ing in  a  force  directed  from  forward  aft.  This  resultant  will 


RESISTANCE. 


79 


i  such  amount  that  the  increased  expenditure  of  energy 
necessary  to  overcome  it  will  just  equal  the  energy  necessary 
to  the  maintenance  of  the  system  of  waves  and  stream-lines. 
Having  thus  referred  to  the  modified  stream-line  system 


and  its  attendant  phenomena  as  a  sufficient  initial  cause  for 
the  formation  of  waves,  we  may  examine  their  actual  charac- 
ter and  distribution  in  more  detail. 

In    Figs.    31,    32,    and    33   are   shown  wave   patterns    as 
determined  by  Mr.  R.  E.  Froude  for  the  various  cases  men- 


8o 


RESISTANCE  AND   PROPULSION   OF  SHIPS. 


tioned.     We  have  here  indicated  two  quite  distinct  varieties 
of  wave  form  and  distribution: 

(a)  Waves  with  crests  perpendicular  to  the  line  of  motion, 

constituting  the  transverse  series; 

(b)  Waves  with  crests  oblique  to  the  line  of  motion,  con- 

stituting the  oblique  or  divergent  series. 

FIQ.  4 

PLANS  OF  WAVE  SYSTEMS  MADE  BY  DIFFERENT  VESSELS 
AT  18  KNOTS  SPEED. 

N.B.  Position  of  Wave  Crests  indicated  by  shading. 
83  FT.  LAUNCH 


FIG.  33- 

The  series  of  transverse  waves  are  distributed  along  the 
line  of  motion  of  the  ship,  and  may  be  considered  as  a  group 


RESISTANCE.  8 1 

of  trochoidal  waves  as  described  in  §  10.  The  speed  of  the 
individual  waves  is  the  same  as  that  of  the  ship,  and  hence 
their  length  from  crest  to  crest  corresponds  closely  to  that 
of  a  deep-water  trochoidal  wave  traveling  at  the  same  speed 
as  the  ship.  A  crest  is  always  found  at  or  near  the  bow,  and 
it  is  this  crest  which  may  be  considered  as  the  initial  result  of 
the  wave-forming  tendency  in  this  region.  To  avoid  confu- 
sion this  crest  is  omitted  from  the  diagram,  and  only  the  more 
pronounced  local  elevations  constituting  the  divergent  crests 
are  shown.  As  we  shall  explain  later,  however,  the  energy 
belonging  to  this  wave  is  drained  away  sternward  relative  to 
the  ship,  and  gives  rise  to  a  series  of  secondary  waves,  or 
echoes  as  they  are  termed,  the  primary  and  its  echoes  thus 
forming  the  series  as  a  whole. 

In  a  long  parallel-sided  ship  the  crests  and  hollows  show 
for  some  distance  from  the  bow,  gradually  decreasing  in 
altitude  as  they  spread  transversely,  and  thus  involve  more 
and  more  water  with  a  gradually  decreasing  energy.  If  the 
parallel  body  is  so  long  that  the  waves  have  by  this  process 
of  dissipation  become  inappreciable,  then  it  will  be  found 
that  a  similar  principal  crest  will  be  generated  near  the  stern, 
the  energy  of  which,  by  the  same  process  of  sternward  propa- 
gation as  at  the  bow,  will  give  rise  to  a  stern  series  of 
echoes.  The  diminution  of  pressure  amidships  will  be  so 
slight  and  will  be  distributed  over  so  great  a  distance  that 
the  primary  hollow  will  usually  be  scarcely  noticeable.  If 
instead  of  the  long  parallel  body  we  have  the  usual  form,  we 
shall  have  the  primary  crest  at  the  bow  and  stern,  and  a 
more  pronounced  tendency  to  form  a  primary  hollow  amid- 
ships. The  energy  of  the  waves  thus  formed  will  be  propa- 
gated sternward  as  before,  thus  giving  rise  to  echoes  and  to 


82  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

three  more  or  less  plainly  marked  initial  systems.  These, 
however,  will  coalesce,  and  the  whole  will  form  a  complete 
system  with  crests  and  hollows  of  size  and  distribution 
depending  on  the  character  of  the  various  components.  As 
the  variations  in  velocity  due  to  the  stream-line  motion  are 
only  slight,  the  altitude  of  such  waves  will  be  small,  and  at 
ordinary  speeds  they  are  scarcely  noticeable.  As  the  increase 
in  velocity  amidships  is,  moreover,  quite  gradual,  the  corre- 
sponding initial  hollow  will  usually  be  slight,  and  a  correspond- 
ingly unimportant  element  in  the  combined  series.  In  many 
cases,  however,  at  high  speeds  and  when  the  elements  of 
combination  enter  properly,  the  combined  series  may  show  a 
very  pronounced  hollow  amidships. 

Turning  to  the  divergent  series,  we  have  the  configuration 
shown  in  the  diagrams  involving  in  each  case  an  initial 
divergent  crest  formed  on  either  side  of  the  bow.  These 
may  be  considered  as  the  primary  result  of  certain  tendencies 
to  which  we  shall  directly  revert.  The  same  drain  of  energy 
and  propagation  sternward  obtains  here  as  with  the  trans- 
verse series,  and,  as  we  shall  explain,  the  result  is  the  series 
of  echoes  as  shown,  arranged  in  skew  or  imbricated  order; 
i.e.,  with  their  crest-lines  overlapping  like  the  shingles  of  a 
house. 

We  have  now  to  explain  the  formation  of  the  primary 
divergent  wave  at  the  bow.  We  have  already  explained  the 
initial  formation  of  the  transverse  crest  at  the  bow.  This 
crest  in  itself  is  of  small  altitude,  and  extends  over  a  consid- 
erable area  according  to  the  speed  and  to  the  rate  of  varia- 
tion of  the  pressure  in  the  stream-lines.  Referring  to  Fig.  34, 
we  have  in  plan  a  suggestion  of  the  contour  or  level  lines  for 
the  variation  of  level  which  would  result  from  the  initial 


RESISTANCE.  83 

impulse  corresponding  to  the  conditions  from  forward  aft. 
The  line  LBHDGFK  is  supposed  to  lie  on  the  surface  of  the 
smooth  water.  AB  and  EF  are  humps  or  crests,  and  CD  is  a 
depression  or  hollow.  Now  the  tendencies  which  lead  to  the 
general  elevation  of  the  water  throughout  AB  become  very 
much  accentuated  near  A,  the  forward  end  of  the  surface  of 
discontinuity  between  the  ship  and  the  liquid.  As  a  result 
there  will  be  raised  against  the  surface  of  the  bow  at  A  a 


FIG.  34. 

more  or  less  pronounced  wall  of  water.  This  may  be  consid- 
ered as  a  locally  exaggerated  manifestation  of  the  general 
tendency  which  would  give  rise  to  the  hump  as  a  whole. 
Otherwise  we  readily  see  in  this  wall  the  simple  result  of 
driving  the  oblique  face  A  rapidly  through  the  water,  or,  vice 
•rcrsa,  the  natural  result  of  placing  an  oblique  face  A  in  a 
rapidly  flowing  stream. 

While  therefore  this  special  elevation  of  water  is  due 
fundamentally  to  the  same  general  cause  as  the  remainder  of 
the  hump,  and  while  it  must  be  considered  as  a  part  of  this 
general  elevation  AB,  yet  it  is  elevated  so  definitely  above 


84  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

the  surface  of  the  remainder  of  the  hump,  that  immediately 
we  find  this  local  elevation  undergoing  propagation  somewhat 
like  a  positive  wave  of  translation.  More  exactly  or  more 
generally,  we  may  view  this  propagation  as  a  result  of  the 
unequal  distribution  of  pressure  in  the  water  in  this  neighbor- 
hood. The  pressure  is  greatest  quite  near  the  ship,  and 
decreases  rapidly  as  we  go  slightly  outward.  The  water 
thus  affected  will  naturally  yield  in  all  directions  in  which 
the  pressure  is  less.  Hence  it  will  yield  upward  and  out- 
ward, and  thug  give  rise  to  an  initial  elevation  of  water  and 
to  a  propagation  outward  in  the  direction  of  the  most  rapid 
decrease  of  pressure.  This  at  first  will  be  nearly  normal  to 
the  water-line  or  surface  of  the  ship. 

Let  us  now  attach  ourselves  to  the  ship  and  consider  the 
stream  as  flowing  by.  Relative  to  the  ship  the  initial  propa- 
gating impulse  will  be  in  the  direction  OP,  sensibly  perpen- 
dicular to  OQ,  Fig.  35.  Let  v  be  the  velocity.  Then  if  at 


FIG.  35- 

a  given  instant  the  stream  could  be  suddenly  arrested,  the 
water  at  that  instant  forming  the  elevation  AB  would  be 
propagated  along  OP  with  velocity  v.  In  the  actual  case, 
however,  the  propagation  takes  place  not  into  still  water,  but 
into  water  moving  sternward  with  its  natural  stream-line 


RESISTANCE.  «5 

velocity,  approximately  parallel  to  AQ.  The  elevation  of 
water  thus  propagated  will  have  two  velocities — one,  v,  along 
OP,  and  another,  w,  along  OQ.  Hence  the  actual  propagation 
will  be  the  resultant  of  these  two,  and  its  direction  OR  will 
make  an  angle  ft  with  OQ  such  that  tan  ft  =  v  —•  w.  Also 
the  direction  OR  will  make  an  angle  y  with  the  direction  of 
motion  AL  such  that 


tan  y  =  tan  (a  -f-  ft)  = 


tan  a  -\ 

1    w 

V 

I  —  —  tan  a 

w 


The  velocity  v  may  be  considered  as  depending  on  the 
rate  of  decrease  of  pressure  from  AQ  outward,  or  on  the 
pressure  very  near  the  surface  of  the  ship  in  this  neighbor- 
hood. We  might  consider  a  portion  of  the  velocity  of 
propagation  as  due  to  the  elevation  AB  considered  as  a 
positive  wave  of  translation.  Inasmuch,  however,  as  the 
distribution  of  pressure  seems  to  be  the  important  factor  in 
this  propagation,  it  seems  preferable  to  refer  the  whole 
phenomenon,  generation  and  propagation,  to  this  general 
cause.  It  is  therefore  evident  that  v  will  increase  and 
decrease  with  the  velocity  of  the  ship  according  to  a  law  too 
complex  for  general  expression. 

The  velocity  w  is  the  velocity  along  the  stream-lines.  It 
will  be  at  this  point  slightly  less  than  u,  the  velocity  of  the 
ship,  or  the  velocity  of  the  flow  as  a  whole.  For  fine  ships 
or  small  values  of  a  its  value  cannot  differ  widely  from 
u  cos  of.  For  large  values  of  a  this  would  give  too  small  a 
value. 

Turning  to  the  value  of  tan  y  above,  it  is  seen  that  the 
only  quantity  depending  on  speed  is  the  ratio  v  -f-  w.  Both 


86 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


of  these,  as  we  have  seen,  depend  on  the  speed  of  the  ship, 
and  it  seems  not  unnatural  to  suppose  that  v  like  w  may  very 
nearly  vary  directly  with  uy  the  speed  of  the  ship.  In  such 
case  v  -T-  w,  and  with  it  yt  will  remain  very  nearly  the  same 
at  varying  speeds.  This  has  been  found  experimentally  to 
be  the  case,  more  especially  for  fine  ships  at  moderate  and 
high  speeds.  For  very  full  ships  the  value  of  y  seems  to 
increase  slightly  with  the  speed,  indicating  that  for  such 
forms  v  increases  more  rapidly  than  w.  Before  leaving  this 
value  of  tan  y  it  may  be  well  to  show  the  method  of  finding 
the  mean  value  of  tan  a  for  the  bow.  In  Fig.  36  let 
the  origin  be  at  B  and  x  reckoned  plus  aft.  Then  if  a 


FIG.  36. 
denotes  the  inclination  of  the  curve  AB  we  have 


tan  a  = 


dx 


i    f*x\  i    PX\         y 

and    mean  of  tan  a  =  —  /      tan  ctdx  =  --  I      dy  =  — . 

X^  l/o  •*",  e/o  Xl 

Integrating  similarly  for  the  mean  vertically,  we  have 

y         i     /* 

mean  of  —  =  -    -  /  y.dz  =  area  of  half-section  AC  -T-  z.x. 
x,       z,x>  J  •* 


Hence 


mean  of  tan  a  for  bow  = 


area  of  half-section  AC 


RESISTANCE. 


In  usual  forms  BC  may  be  taken  at  10  to  15  per  cent  of 
the  length. 

At  very  high  velocities  of  the  ship  the  water  in  the  eleva- 
tion AB,  Fig.  35,  instead  of  forming  a  simple  sharply-marked 
mound  will  curl  and  break.  The  general  result  will  be  the 
same,  however,  as  the  breaking  is  simply  equivalent  to  pour- 
ing a  mass  of  water  along  AB,  and  the  distribution  and  rate 
of  variation  of  the  pressure  will  be  such  as  to  give  the  same 
general  outward  propagation  as  above  described. 

Having  thus  discussed  the  propagation  of  a  single  instan- 
taneous elevation  of  water  AB  and  the  considerations  on  which 
its  direction  depends,  we  next  observe  that  as  the  propaga- 
tion removes  the  water  forming  the  elevation  at  any  one 
instant,  it  is  immediately  succeeded  by  another  like  elevation, 
so  that  as  a  result  there  is  a  continual  elevation  propagated 
relative  to  the  boat  along  the  direction  OR.  As  the  eleva- 
tion recedes  from  AQ,  however,  the  crest  tends  to  broaden  and 
thus  loses  altitude  and  finally  becomes  insensible.  Toward 
the  outer  end  its  direction  may  change  somewhat,  due  to  the 
superposition  of  the  tendencies  at  that  point  on  the  initial 
impulse  along  OP.  As  the  side  of  the  ship  is  left,  the  most 
rapid  decrease  of  pressure  will  be  directed  more  nearly  trans- 
versely, or  even  aft,  toward  the  hollow  amidships.  In  con- 
sequence of  this,  together  with  the  changed  value  of  w,  the 
outer  end  may  curve  around  somewhat  toward  the  stern. 
Due,  however,  to  the  decreasing  altitude,  this  change  is 
usually  unnoticeable. 

We  have  thus  described  the  development  of  the  primary 
member  of  the  divergent  system  of  waves  at  the  bow.  A 
similar  primary  is  formed  at  the  stern  through  the  action  of 
the  same  general  causes.  In  this  case  the  initial  wall  or 


88 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


special  elevation  of  water  is  formed  under  the  stern  and  is 
propagated  outward  into  the  outlying  water,  which,  due  to 
the  frictional  wake  and  to  the  stream-line  motion  in  this 
neighborhood,  has  a  somewhat  lower  velocity  than  at  the 
bow.  While  thus  the  initial  direction  of  propagation  of  the 
elevation  would  usually  be  somewhat  aft  of  the  direction  for 
the  similar  elevation  at  the  stern,  yet  the  other  component 
being  less  than  at  the  bow  the  resulting  direction  is  not  far 
different  at  the  two  ends.  At  high  speeds  instead  of  the 
special  elevation  being  formed  against  the  side  of  the  ship  it 
is  formed  immediately  astern.  In  this  case  the  two  streams 
flowing  one  on  either  side  of  the  ship  combined  with  that 
flowing  under  the  ship  may  be  considered  as  meeting  and 
giving  rise  to  an  initial  elevation  along  the  line  AB,  Fig.  37. 


FIG   37- 

This  being  propagated  transversely  outward  on  each  side  into 
the  outlying  water  gives  rise  as  before  to  the  two  divergent 
waves  ABP  and  ABQ.  At  certain  speeds  and  with  certain 
forms  of  boat  this  wave  is  very  pronounced,  taking  the  form 
of  a  flat-topped  hill  of  water  sloping  away  in  a  fan-shaped 
figure  and  gradually  decreasing  in  altitude.  At  still  higher 
speeds  this  wave  recedes  still  farther  astern  and  usually 


RESISTANCE.  »9 

becomes  somewhat  less  pronounced.  Especially  is  this  the 
case  with  that  form  of  after-body  which  approximates  to  a 
wedge  with  flat  side  down  and  edge  near  the  water  surface  at 
the  stern,  as  found  on  most  modern  torpedo-boats  and  many 
other  high-speed  craft. 

Having  thus  discussed  the  formation  of  the  primary 
members  of  the  various  series  of  waves  involved  in  the 
problem  of  resistance,  it  is  time  to  take  up  the  question  of 
the  propagation  of  energy  already  referred  to. 

12.  THE  PROPAGATION  OF  A  TRAIN  OF  WAVES. 
This   subject   has   been   briefly  considered  in  §  10.     We 
must  now  examine  in  more  detail  the  consequences  of  the 
statements  there  made.     In  Fig.  38  let  the  full  line  represent 


C, 


\s 

FIG.  38. 


a  group  of  waves  of  which  C  marks  the  center  and  principal 
member,  the  successive  crests  on  either  side  gradually 
decreasing  in  altitude  until  they  become  negligible.  Then 
observation  in  agreement  with  theory  shows  that  such  a  con- 
figuration or  group  is  propagated  with  only  about  one  half 
the  velocity  of  the  individual  waves  A,  B,  C,  etc.  Now  the 
group  as  a  whole  may  be  considered  as  the  expression  of  a 
certain  distribution  of  energy,  and  hence  the  velocity  of  the 
group  may  be  considered  as  the  velocity  with  which  the 
energy  is  propagated,  as  distinguished  from  the  velocity  of 
the  individual  waves.  It  must  be  considered,  therefore,  that 
in  waves  of  trochoidal  character  the  energy  is  propagated  at 
only  one  half  the  velocity  of  the  waves  themselves.  Viewin 


QO  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

the  matter  from  this  standpoint,  let  us  examine  the  propaga- 
tion of  the  group  of  waves  in  Fig.  38.  The  wave  C  repre- 
senting the  principal  member  is  propagated  forward  with  a 
certain  velocity.  As  it  moves  forward,  however,  it  leaves 
behind  a  part  of  its  energy,  and  hence  gradually  decreases  in 
altitude,  and  is  no  longer  the  maximum  member  of  the 
group.  The  energy  thus  left  behind  is  absorbed  by  B,  which 
therefore  increases  and  becomes  after  a  certain  time  the 
maximum  member  Blt  and  hence  the  new  center  of  the 
group.  The  wave  C  in  the  meantime  has  gone  from  C  to  C^ 
while  the  principal  member  of  the  group,  as  such,  has  gone 
from  C  to  B^  The  latter  evidently  fixes  the  velocity  of  the 
group.  There  is  thus  a  continual  propagation  of  the  indi- 
vidual waves  forward,  while  relative  to  the  waves  the  indi- 
vidual characteristics  are  propagated  backward.  The  group 
velocity  is  therefore  the  difference  between  these  two.  It 
results  that  any  individual  wave  as  A  travels  along  through 
the  entire  group,  taking  on  successively  the  characteristics  of 
the  various  members,  and  finally  dying  out  at  the  forward 
edge  while  a  new  member  rises  at  the  after  edge;  and  thus  the 
process  continues.  Or,  again,  we  may  consider  that  the  lead- 
ing waves  are  continually  dwindling  and  disappearing  for  lack 
of  the  energy  which  they  leave  behind,  while  the  following 
waves  similarly  grow  by  the  access  of  energy  which  they  as 
continually  receive.  Thus  if  v  is  the  velocity  of  the  waves 
forward,  then  the  velocity  of  the  group  and  of  the  energy  is 
v/2  forward.  Relative  to  the  group  the  velocity  of  the 
waves  is  v/2  forward,  and  of  the  energy  o.  Relative  to  the 
waves  the  velocity  of  the  group  and  of  the  energy  is  v/2 
backward. 

In  thus  considering  the  group  velocity  of  a  train  of  waves 


RESISTANCE.  gi 

we  must  note  that  the  exact  proof  supposes  trochoidal 
motion,  and  hence  circular  paths  for  the  motion  of  the  parti- 
cles. This  cannot  be  the  case  where  the  waves  vary  in 
altitude,  and  under  such  circumstances  the  group  cannot  be 
indefinitely  permanent,  even  neglecting  viscous  degradation. 
This  is  borne  out  by  observation.  Again,  in  shallow  water 
the  orbits  being  oval  or  approximately  elliptical,  the  group- 
velocity  is  greater  than  one  half  the  wave-velocity,  and  hence 
in  such  case  the  energy  is  transmitted  at  a  velocity  greater 
than  one  half  that  of  the  individual  waves.  In  the  extreme 
case  we  have  the  wave  of  translation  in  which  the  group 
becomes  reduced  to  the  individual,  the  velocity  of  one  being 
necessarily  that  of  the  other.  In  such  case,  then,  the  energy 
is  transmitted  at  a  velocity  equal  to  that  of  the  individual 
wave. 


13.  FORMATION  OF  THE  ECHOES  IN  THE  TRANSVERSE  AND 
DIVERGENT  SYSTEMS  OF  WAVES. 

Taking  first  the  primary  bow  transverse  wave,  we  see  that 
as  an  individual  it  must  necessarily  keep  pace  with  the  ship. 
Considering,  however,  that  it  is  constituted  approximately  as 
a  trochoidal  wave,  its  energy  will  be  continually  draining  to 
the  rear  and  giving  rise  to  a  successive  series  of  echoes. 
These  from  their  gradual  spreading  sideways  soon  lose  all 
significant  altitude  and  become  negligible. 

Turning  now  to  the  divergent  system,  we  find  in  the 
draining  backward  of  the  energy  relative  to  the  ship  com- 
bined with  the  natural  internal  propagation  peculiar  to  this 
wave,  the  explanation  of  the  overlapping  or  skew  arrange- 
ment of  the  crests.  We  may  consider  that  these  divergent 


92  RESISTANCE  AND  PROPULSION  OF  SHIPS. 

waves  constitute  virtually  a  special  series  superimposed,  as  it 
were,  on  the  flatter  transverse  series.  In  Fig.  39  let  AB  be 
the  primary  member.  Now  this  crest,  considered  simply  as 
a  configuration,  is  by  virtue  of  its  fixed  relation  to  the  ship 
carried  forward  unchanged  in  the  direction  OP  with  a 
velocity  v  equal  to  that  of  the  ship.  A  part  of  the  energy, 
however,  will  naturally  lag  behind,  the  relative  amount 
approximating  more  or  less  closely  to  one  half,  according  as 
the  characteristics  of  the  wave  approach  more  or  less  closely 


FIG.  39. 

to  those  of  trochoidal  form.  Relative  to  still  water,  therefore, 
there  will  be  a  propagation  of  the  energy  forward  at  a 
velocity  less  than  v,  and  approaching  v/2  as  the  wave 
approaches  trochoidal  form.  In  addition  to  this  propagation 
of  energy  along  OP,  there  will  be  likewise,  due  to  the 
peculiar  constitution  of  this  elevation  of  water,  a  propagation 
or  transmission  of  energy  outward  along  OQ.  The  resultant 
direction  of  propagation  is  therefore  along  some  line  OR. 
The  result  is  that  the  energy  in  AB  may  be  considered  as 
located  at  a  later  instant  in  AtBlt  while  the  primary  crest  at 


RESISTANCE.  93 

A ' B'  exists  by  virtue  of  energy  drawn  from  the  ship  between 
A  and  A'.  Or  otherwise,  relative  to  the  ship  and  the  con- 
stantly renewed  primary  crest  AB,  energy  is  constantly  drain- 
ing to  the  rear  and  outward  along  OQ,  the  result  being  a 
transmission  obliquely  outward  in  such  manner  as  to  give  rise 
to  the  series  of  overlapping  crest-lines  as  shown. 

In  a  similar  way  a  second  echo  is  formed,  and  so  on  until 
by  lateral  extension  and  viscous  degradation  the  altitude 
becomes  negligible.  In  like  manner  the  stern  series  of 
divergent  waves  may  give  rise  to  a  similar  series  of  echoes. 

We  have  already  referred  to  Figs.  32  and  33  in  illustra- 
tion of  the  wave-pattern  whose  formation  we  have  attempted 
to  explain.  We  may  now  call  attention  to  certain  further 
details. 

We  first  note  the  location  of  the  primary  bow  and  stern 
waves.  At  low  speeds  the  bow  divergent  waves  are  formed 
close  to  the  bow,  while  as  the  speed  increases  their  mean  loca- 
tion draws  somewhat  farther  aft.  At  low  speeds  the  stern 
divergent  waves  are  formed  on  the  quarter  and  diverge 
independently  as  shown  in  Fig.  32.  As  the  speed  increases 
they  draw  aft  and  together,  and  finally  coalesce  into  an  elon- 
gated mound  of  water  spreading  away  in  V  shape,  as  already 
noted. 

The  bow  and  stern  transverse  primary  waves  have  the 
same  general  location  longitudinally  as  the  divergent  pri- 
maries. Indeed,  as  we  have  already  pointed  out,  the  latter 
are  merely  local  exaggerations  of  the  general  primary  eleva- 
tion at  these  points. 

The  length  between  the  primary  bow  and  stern  waves 
increases  somewhat  with  the  speed.  At  moderate  or  high 
speeds  it  is  usually  considered  as  slightly  greater  than  the 


94  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

length  of  the  ship.  The  available  data  is  not  sufficient  to 
determine  satisfactorily  the  amount  of  excess,  but  it  is  usually 
taken  at  from  5  to  10  per  cent  of  the  length. 

Numerous  measurements  of  the  wave-lengths  for  the 
transverse  system  are  found  to  be  in  close  accord  with  the 
length  for  the  natural  trochoidal  wave  of  the  same  speed  as 
the  ship,  and  as  given  by  formula,  §  10  (i).  It  may  therefore 
be  considered  as  shown  by  experience  that  the  wave-length 
of  the  transverse  series  is  in  satisfactory  agreement  with  what 
it  should  be,  considering  them  as  a  series  of  trochoidal  waves 
having  the  speed  of  the  ship.  In  consequence  it  seems 
allowable  to  assume  for  the  waves  of  this  system  such  further 
properties  of  trochoidal  waves  as  may  be  convenient  for  their 
investigation. 

With  the  divergent  system  the  case  is  somewhat  different. 
These  waves,  so  far  as  their  configuration  is  concerned,  travel 
forward  with  the  velocity  of  the  ship.  If  y  is  the  angle  of 
their  crest  with  the  longitudinal,  however,  the  component  of 
their  velocity  normal  to  the  crest-line  will  be  u  sin  y.  Now 
measurement  indicates  that  the  wave-length  in  this  direction 
agrees  fairly  well  with  that  of  a  natural  trochoidal  wave 
having  a  velocity  u  sin  y.  It  follows  that  in  a  sense  we  may 
consider  this  system  as  made  up  of  a  series  of  bits  of  tro- 
choidal waves  with  a  velocity  u  sin  y  and  a  length  corres- 
ponding. For  this  length  we  shall  have 

27T 

L  =  — z/2  sin2  y. 
g 

Hence  for  the  length  from  crest  to  crest  on  a  longitudinal 
line  we  should  have 

Zj  =  — u*  sin  y. 


RESISTANCE.  95 

For  the  transverse  system  we  have  likewise 


Hence  the  longitudinal  length  of  the  transverse  series  should 
be  somewhat  greater  than  for  the  divergent  series.  This  is 
borne  out  by  the  diagrams,  reference  to  which  shows  that  the 
crests  of  the  former  fall  farther  and  farther  astern  relative  to 
those  of  the  latter.  While  the  difference  may  not  be  so 
great  as  is  indicated  by  the  formulae  above,  the  tendency  is 
in  this  direction,  and  the  errors  of  the  formulae  may  naturally 
be  referred  to  the  imperfect  manner  in  which  such  waves 
fulfil  the  trochoidal  characteristics. 


14.  WAVE-MAKING  RESISTANCE. 

Having  thus  considered  the  formation  of  the  various  wave 
systems  attending  the  motion  of  a  ship  through  the  water, 
we  may  next  examine  their  relation  to  resistance. 

Taking  first  the  bow  transverse  system,  we  have  seen  that 
the  energy  of  the  waves  is,  relative  to  the  ship,  constantly 
passing  sternward.  At  the  same  time  the  energy  of  the 
system  as  a  whole  must  be  maintained  constant.  Hence  this 
drain  of  energy  sternward  must  be  made  up  by  energy 
derived  from  the  ship,  and  it  is  simply  the  transmission  of 
this  energy  which  gives  rise  to  this  part  of  the  wave  resist- 
ance. The  exact  fraction  of  energy  which  falls  astern  rela- 
tive to  the  ship  will  depend  on  the  geometrical  character  of 
the  wave;  and  it  must  not  be  considered  that  the  ratio  1/2 
is  anything  more  than  an  approximation,  which,  however, 
may  serve  for  illustrative  purposes.  Hence  let  us  assume 


96  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

that  one  half  the  energy  is  naturally  propagated  with  the 
form,  and  that  one  half  must  be  made  up  from  the  ship.  It 
follows  that  for  every  two  wave-lengths  run  the  ship  must 
supply  the  energy  necessary  to  the  formation  of  one  wave. 

Let  E  be  the  energy  of  the  wave,  L  the  wave-length,  and 
R  the  resistance  due  to  its  maintenance.  Then  since  resist- 
ance equals  the  work  done  or  energy  transmitted  divided  by 
the  distance,  we  have 

R=—T     or     R~j. 
2L  L 

A  similar  expression  would  hold  for  the  element  of  resist- 
ance due  to  each  of  the  divergent  waves  at  the  bow,  E  being 
the  energy  and  L  the  longitudinal  wave-length  for  this 
system. 

It  thus  appears  that  the  work  which  is  done  by  the  ship 
at  the  bow  on  the  water  in  maintaining  these  systems  of 
waves  is  equivalent  to  adding  one  new  wave  to  each  system 
for  every  two  wave-lengths  traveled.  Similar  considerations 
hold  for  the  resistance  due  to  the  stern  wave  systems. 

We  have  thus  far  considered  the  bow  and  stern  wave 
systems  in  their  individual  aspect.  We  have  now  to  consider 
the  possible  influence  of  the  former  upon  the  latter. 

We  first  note  that,  so  far  as  the  divergent  system  is  con- 
cerned, no  direct  influence  is  possible,  since  these  waves 
travel  off  obliquely  and  come  in  contact  with  the  ship  at  the 
bow  only.  Again,  if  the  ship  is  very  long  for  her  speed, 
especially  if  she  has  a  long  middle  body,  the  bow  transverse 
system  will  become  negligible  before  reaching  the  stern,  so 
that  each  system  will  produce  its  individual  effect.  In  the 
general  case,  however,  the  bow  transverse  system  will  not 


RESISTANCE. 


97 


have  entirely  disappeared  by  the  time  it  reaches  the  stern, 
and  there  will  be  formed  at  this  point  a  combination  system 
resulting  from  the  remainder  of  the  bow  system  and  the 
natural  stern  system. 

The  total  energy  of  the  entire  wave  systems  may  be  con- 
sidered as  that  of  the  bow  transverse  and  divergent  systems 
plus  that  of  the  stern  divergent  system,  possibly  modified  by 
the  bow  transverse  system,  plus  that  of  the  stern  combina- 
tion system  minus  that  part  of  the  latter  which  is  received 
from  the  bow  system.  For  the  transverse  systems  alone  the 
above  value  is  equal  to  the  sum  of  the  energy  of  the  stern 
combination  system  plus  the  loss  of  energy  in  the  bow  system 
when  it  reaches  the  stern. 

Let  L  denote  the  distance  between  the  natural  bow  and 
stern  primary  crests  as  discussed  in  §  13,  and  let  A  be  the 
wave-length,  which  we  may  take  as  that  of  a  trochoidal  wave 
of  the  velocity  of  the  ship.  Let  L  =  n\  -\-  a.  Then  a  is 
the  distance  from  the  stern  primary  crest  forward  to  the 
nearest  crest  of  the  bow  system.  Hence  a  -=-  ^  is  the  phase 
difference  ratio  and  2^a  -~  A  is  the  phase  difference  angle. 
Let  ^,  be  the  altitude  of  the  bow  wave,  kh^  the  remaining 
altitude  of  the  bow  system  when  it  reaches  the  stern,  and  //3 
the  natural  altitude  of  the  stern  wave.  Then  referring  to 
§  10  (15)  it  is  seen  that  the  combination  system  will  be  tro- 
choidal and  of  altitude 


=  p/i?  +  h?  + 


271(1 


COS 


Referring  now  to  §  10  (i  i),  it  is  seen  that  the  energy  of  a 
wave  p?r  unit  breadth  measured  along  the  crest  is  propor- 
tional to  the  product  of  the  wave-length  by  the  square  of  the 


98  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

altitude.  Let  Bl  be  the  breadth  transversely  at  the  bow. 
This  is  of  course  indefinite,  since  //,  gradually  decreases  as  we 
go  from  the  bow  outward.  We  may,  however,  consider  h* 
as  the  mean  of  the  squares  of  the  altitudes  for  a  breadth 
Bl  considered  as  comprising  all  of  the  wave  whose  elevation 
is  sensible.  The  energy  of  the  bow  primary  wave  will  then 
be  proportional  to  h?B^.  Let  B  and  h  denote  similar  quan- 
tities for  the  combination  system  at  the  stern.  Then  the 
energy  of  the  stern  combination  wave  will  be  proportional  to 
h*Bh.  A  portion  of  the  latter,  however,  proportional  to 
fzhiBj^,  is  derived  from  the  bow  system.  Hence  the  energy 
necessary  for  the  maintenance  of  the  whole  system,  and  to 
be  supplied  by  the  ship  in  running  a  distance  2A,  is  propor- 
tional to  h?B^  +  tfB^  —  /P/V^A.  Now  assuming  that  Bl 
and  B  are  sensibly  the  same,  substituting  for  the  value  of  // 
above  and  putting  the  result  proportional  to  the  work  done 
by  the  ship  in  overcoming  a  resistance  Rm  through  a  distance 
2A,  we  have 


(h?  +  h?  +  2khji,  cos  ^ 


Whence          Rw  ~  (/;,'  +  h?  +  zkhji*  cos  —-B.      .      .     (i) 


Let  s  =  mL  =  wave-making  length  of  ship,  in  which,  as 
already  noted,  m  is  usually  1.05  to  i.io.  Then  s  will  consist 
of  a  certain  number  of  multiples  of  A  with  a  remainder  a. 
Hence  cos  (2ns  -f-  A)  —  cos  (2na  -=-  A).  Also,  A  =  2nv*  ~-  g, 
§  10  (i).  Hence  2ns  -=-  A  —  gs  ~  v1—  gmL  ~  v1,  and  hence 


h?  +  2khjt,  cos 


RESISTANCE. 
Now  in  general  we  may  put 


99 


and 

and  hence 


~~  v*, 


h*  ~  //a2 


Hence  we  may  represent  the  resistance  due  to  the  natural 
bow  system  by  a  term  of  the  form  //"V,  and  that  due  to  the 
natural  stern  system  by  a  similar  term,  /  V.  Then  from  the 
derivation  of  (2)  it  follows  that  we  should  likewise  represent 
the  total  resistance  of  the  combined  series  by  an  expression 
of  the  form 

.     .     .     (3) 


This  expression  is  periodic  in  character,  the  mean  value 
being  (H*  +  J*)Bv\     To  illustrate  the  variation   from    this 


10   12   14   10   18   20   22   24   26 


30   33   34   36   38   40 


Values  of  iHJk  cos       ^laid  off  from  OX  as  axis. 

v1 

FIG.  40. 

value  due  to  the  third  term  we  may  refer  to  Fig.  40.     We 
have  here  illustrated  the  values  of  the  expression 


2HJk 


V 


using  the  following  numerical  values-  of  the  terms 

2HJk=   I, 

mL  =  100. 


TOO 


RESISTANCE   AND    PROPULSION  OF  SHIPS. 


At  low  and  moderate  speeds  the  amount  of  wave-resist- 
.ance  is  so  small  that  these  fluctuations  are  entirely  imper- 
ceptible. As  the  speed  is  increased,  however,  a  point  is 
reached  where  the  wave-resistance  forms  an  important  part 
of  the  total  resistance,  and  where  evidences  of  such  a 
periodic  fluctuation  in  its  amount  are  plainly  indicated.  This 
is  illustrated  in  Fig.  41,*  showing  the  residual  resistance- 


10       11 


13 


13 


14        15        10        17        18        19        20 

SPEED  IN   KNOTS 
CURVES  OF  RESIDUAL  RESISTANCE 


21 


22        23 


Ship 

Length 

Beam 

Draft! 

Disp't 

A 

400' 

38.2 

20.7 

5930 

D 

* 

*• 

19.1 

5390 

C 

»• 

" 

16.5 

4480 

D 

»• 

15.4 

4090 

FIG.  41. 

curves  of  a  ship  at  sexeral  drafts  as  determined  by  Mr.  R.  E. 
Froude, 

For  the  waves  of  the  divergent  series  we  may  take  like- 
wise the   energy  as   proportional  to   the   length,  breadth,  and 


*  Transactions  Institute  of  Naval  Architects,  vol.  xxil.  p.  220. 
\  Inclusive  of  9  inches  of  keel. 


RESISTANCE.  IOI 

square  of  the  altitude.  Hence  //,  B,  and  A  denoting  in 
general  the  same  characteristics  as  above,  we  should  have  for 
each  train 

Energy  ~  J?B\, 
R  ~  J?B. 

As  further  illustrating  the  above  principles,  we  may  refer 
to  the  following  experiments  of  Mr.  Froude  on  the  influence 
of  length  of  parallel  middle  body  on  wave-making  resistance. 

For  the  purposes  of  the  investigation,  an  initial  model  was 
taken  as  representing  a  ship  160  feet  long.  Successive 
lengths  of  parallel  midd'e  body  representing  20  feet  in  length 
were  then  added  until  the  total  length  represented  500  feet. 
These  models  were  tried  at  various  speeds  and  the  resistance 
measured.  The  frictional  resistance  being  then  computed 
and  subtracted,  the  residual  resistance  was  considered  as 
sensibly  due  to  the  maintenance  of  the  wave  systems.  In 
order  to  show  the  results  graphically,  the  information  was 
disposed  as  in  Fig.  42.* 

Any  given  curve  above  AA  represents  the  residual  resist- 
ance at  a  constant  speed,  as  rmrked,  for  varying  values  of  L 
as  shown  on  the  base  A  A.  Similarly  the  corresponding  line 
below  AA  shows  the  frictional  resistance  at  the  same  speed 
for  the  same  series  of  ships,  the  ordinates  in  this  case  being 
measured  downwards. 

We  may  note  the  following  points  indicated  by  this 
diagram: 

(i)  The  curves  of  residual  resistance  show  plainly  a 
periodic  variation,  the  length  of  each  period  or  distance  from 

*  Transactions  Institute  of  Naval  Architects,  vol.  XVIII.  p.  77. 


102  RESISTANCE   AND    PROPULSION  OF  SHIPS. 


maximum  to  maximum  being  approximately  constant  at  any 
one  speed. 

(2)  The  length  of  the  period  or  spacing  between  maxima 


RESISTANCE  IN  TONS 


RESISTANCE  IN  TONS 


g  S 


I       I       I       I       I    ecg.  I  ,1.1       I       I  | 


I      I     I      I      I        " 

H         E        5 

RESISTANCE  IN  TONS  RESISTANCE  IN  TONS 

FIG.  42. 

is  greater  as  the  speed  increases,  the  value  being  found  to 
vary  nearly  as  the  square  of  the  speed. 

(3)  The  amplitude  of  the  variation  increases  as  the  speed 
increases. 


RESISTANCE.  103 

(4)  The  amplitude  of  the  variation  decreases  as  the  length 
increases. 

The  increase  of  (3)  is  due  to  the  rapidly  increasing  im- 
portance of  wave-making  resistance  in  general,  after  passing 
a  certain  speed,  and  also  more  directly  to  the  value  of  k, 
which  usually  increases  with  the  speed.  The  decrease  of  (4) 
is  due  to  the  decrease  in  k  as  L  increases.  If  L  were  suffi- 
ciently great  there  would  be  no  interference,  and  hence  no 
fluctuation  or  periodic  variation  in  the  resistance. 

Turning  back  now  to  formula  (2)  we  may  further  examine 
the  relation  of  h^  li^,  and  B  to  the  speed. 

In  perfect  stream-line  motion  in  horizontal  planes  and 
without  elevation  of  the  surface  we  have,  denoting  the 
velocity  by  u, 

p  -  /„  =  ^-(U;  -  «')•     See  §2  (i) 

Hence  along  any  one  line  for  a  slight  change  in  velocity 

cr 
dp  =  —  —udu. 

If  now  we  put  du  =  eu,  where  e  represents  a  certain  frac- 
tion, we  have 


Hence  with  varying  initial  velocity,  the  pressure  incre- 
ments or  decrements  at  any  given  point  will  vary  as  the 
square  of  the  velocity.  In  actual  wave-line  motion  at 
moderate  speeds  the  elevation  will  correspond  nearly  to  the 
increment  of  pressure,  and  hence  we  should  find  the  values  of 
//,  and  /*,  nearly  proportional  to  «*.  At  higher  speeds  the 
stream-line  motion  is  less  perfect,  and  it  seems  probable  that 


104  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

the  variation  of  pressure  for  a  given  percentage  of  speed 
fluctuation  will  increase  somewhat  more  slowly  than  as  the 
square  of  the  speed.  Assuming,  however,  in  general  this 
relationship,  we  may  as  before  represent  the  resistance  due  to 
the  natural  bow  and  stern  waves  by  //V  and  /  V.  We 
shall  then  have  again  as  in  (3) 

•\-2kHJ 

Next  as  to  B.  It  has  been  usually  assumed  that  B  is 
very  nearly  constant  at  varying  speeds,  and  that  it  may  be 
taken  as  approximately  proportional  to  the  linear  dimensions, 
of  the  ship.  In  such  case  we  should  have 


Some  have  thought  that  it  should  be  considered  rather  as 
varying  with  ^,  and  hence  with  &2.  In  such  case  we  should 
have 

R»  ~  «', 

and  hence  independent  directly  of  the  dimensions  of  the 
ship.  Risbec  *  suggests  a  combination  of  these  two  terms  in 
the  form 


and  believes  that  as  u  increases  B  will  become  more  and 
more  predominant  in  the  value  of  Rw.  It  may  be  noted  that 
Risbec's  suggestion  is  equivalent  to 


in  which  J  may  depend  on  the  dimensions  of  the  ship  and  n 
may  vary  from  4  to  6. 


*  Bulletin  de  1'Association  Technique  Maritime,  vol.  v.  p.  52. 


RESISTANCE. 


105 


Similar  considerations  hold  with  regard  to  the  resistance 
due  to  the  divergent  system,  each  member  of  which  may  be 
considered  of  the  form 


R  =  Cu4 


or 


or  as  a  summation  of  such  terms,  according  as  the  law  is 
assumed  to  be  constituted.  Hence  we  may  consider  that  in 
reference  to  speed  the  total  wave-making  resistance  will 
probably  vary  with  some  index  between  4  and  6,  or  as  a  sum 
of  terms  with  these  or  intermediate  exponents.  The  prob- 
ably decreasing  values  of  P  and  Q  with  increasing  speed 
would,  if  they  were  taken  as  constant,  decrease  the  exponents 
of  u  from  the  values  they  would  otherwise  have,  and  thus 
perhaps  partly  compensate  for  the  increase  in  exponent  due 
to  the  variation  of  the  term  B. 

The  actual  relation,  however,  of  the  total  wave-making 
resistance  to  the  speed  and  to  the  dimensions  of  the  ship  is 
not  known,  nor  can  it  be  satisfactorily  determined  from  the 
data  available.  There  seems  to  be  ground,  however,  for  the 
belief  that  in  many  cases  it  shows  a  higher  rate  of  variation 
than  that  given  by  the  exponent  4.  See  also  §  26.  For  the 
present,  therefore,  we  are  thrown  back  on  more  empirical 
formulae,  intended  simply  to  express  as  nearly  as  may  be  the 
results  of  experience.  Such  formulae  will  be  found  in  §  29. 
With  the  accumulation  of  data  from  model  experiments  we 
may  perhaps  hope  ultimately  for  a  satisfactory  determination 
of  the  constants  in  (i)  and  (2)  or  in  other  like  formulae  shown 
by  the  data  to  be  more  suitable  to  the  purpose  in  view. 


io6 


RESISTANCE  AND   PROPULSION   OF  SHIPS. 


15.  RELATION  OF  RESISTANCE  IN  GENERAL  TO  THE 
DENSITY  OF  THE  LIQUID. 

For  the  head-resistance  and  skin-resistance  per  unit  area 
we  have  seen  that  the  density  of  the  liquid  enters  as  a  factor. 
Similarly  here  the  density  must  be  a  factor  of  the  energy  of 
a  wave,  and  thus  all  parts  of  the  resistance,  and  hence  resist- 
ance as  a  whole,  will  vary  with  the  density.  The  variation 
in  the  resistance  of  a  given  ship  with  change  of  density  of  the 
liquid  is,  however,  in  large  measure  offset  by  the  oppositely 
varying  amount  of  immersed  body  of  ship.  As  between 
fresh  and  salt  water  the  difference  is  slight,  and  is  usually 
neglected  except  in  the  case  of  model  experiments  or  very 
careful  estimates.  The  densities  of  fresh  and  salt  water  are 
usually  taken  in  the  ratio  of  i.o:  1.032,  or  less  accurately  in 
the  ratio  35  :  36. 

16.  FROUDE'S  EXPERIMENTS  ON  FOUR  MODELS. 

In  1876  Wm.  Froude  published  a  paper*  showing  the 
results  of  experiments  on  the  comparative  resistance  of 
models  of  four  ships  of  the  same  displacement  but  of  certain 
differences  of  form.  These  models  were  called  A,  B,  C,  D\ 
A  being  that  of  the  Merkara.  The  dimensions  correspond- 
ing to  the  four  models  were  as  follows: 


1 

Displace- 
ment. 

Length  of 
Fore 
Body. 

Length  of 
Middle 
Body. 

Length  of 
After 
Body. 

Total 
Length. 

Beam. 

Draft. 

Wetted 
Surface. 

A 

i       r 

144 

72 

144 

360 

37-2 

16.25 

IS  660 

B 

i       i 

179-5 

179-5 

359 

45-88 

18 

19130 

C 

i 

154-5 

154-5 

309 

49-4 

19.32 

17810 

D 

)       i 

95 

95 

95 

285 

45.56 

17.89 

16950 

*  Transactions  Institute  of  Naval  Architects,  vol.  xvu.  p.  181. 


RESISTANCE.  IO/ 

The  forms  of  the  fore  and  after  bodies  of  B,  C,  and  D 
were  derived  from  those  of  A  by  expansion  longitudinally, 
transversely,  and  vertically  as  required.  The  fore  and  after 
bodies  themselves  have  therefore  the  same  ratios  of  fullness 
and  other  characteristics  as  those  of  A,  and  the  resulting 
differences  in  resistance  may  therefore  be  considered  as  due 
to  the  effect  of  the  presence  or  absence  of  the  parallel  middle 
body. 

The  results  may  be  summarized  by  saying  that  at  all 
speeds  B  and  C  had  less  resistance  than  A  or  D.  As 
between  B  and  C,  the  latter  having  the  less  wetted  surface 
had  the  less  resistance.  As  between  A  and  D,  the  latter, 
though  having  less  wetted  surface,  had  at  the  speeds  tried 
(from  9  knots  upward)  greater  resistance,  though  at  the 
lowest  speeds  the  directions  of  the  two  resistance-curves  were 
such  as  to  indicate  an  intersection,  and  lower  values  for  A  for 
very  low  speeds.  At  high  speeds  B  and  C  changed  places 
relatively,  the  smaller  wave-resistance  due  to  B's  greater 
ratio  of  L  to  B  being  more  influential  at  such  speeds  than  its 
greater  wetted  surface.  At  all  speeds  A  had  less  resistance 
than  D.  For  the  former  the  resistance  began  to  increase 
rapidly  at  about  17  knots  and  for  the  latter  at  about  14 
knots.  For  C  this  relatively  rapid  increase  in  resistance 
began  to  appear  at  about  19  knots,  while  for  no  speed  up  to 
2O  knots  was  such  tendency  noticeable  with  B. 

It  appears  therefore  in  general  that  resistance  will  be 
decreased  if,  instead  of  a  parallel  middle  body,  the  fore  and 
after  bodies  are  expanded  so  as  to  obtain  a  ship  of  the  same 
displacement  as  with  parallel  middle  body,  even  if  the  ratio  B 
to  L  is  thereby  somewhat  increased. 


108        RESISTANCE  AND  PROPULSION  OF  SHIPS. 

17.  MODIFICATION  OF  RESISTANCE  DUE  TO  IRREGULAR 

MOVEMENT. 

It  is  well  known  experimentally  in  all  cases  of  bodies 
moving  in  a  liquid,  that  a  certain  amount  of  water  is 
momentarily  bound  to  the  body  and  must  be  considered  as 
practically  taking  part  in  its  motion,  especially  in  any  rapid 
changes  involving  accelerations  and  retardations-  This 
amount,  it  also  appears,  may  be  from  15  to  20  per  cent  of 
the  displacement  of  the  ship.  It  follows  that  the  resistance 
to  an  acceleration  will  not  be  that  due  simply  to  the  mass  of 
the  ship,  but  rather  to  the  combined  mass  of  ship  and  water. 
At  the  same  time  it  must  be  remembered  that  during  retar- 
dation a  corresponding  effort  is  given  out.  It  is,  however,  a 
general  principle  that  more  energy  is  required  to  maintain  a 
system  in  irregular  movement,  the  velocity  oscillating  about 
a  mean  value  u,  than  at  the  same  uniform  value  of  the 
velocity.  This  is  because  in  general  R  varies  with  u  accord- 
ing to  a  higher  index  than  I,  and  because  in  such  case  the 
mean  of  a  series  of  such  powers  of  u  is  greater  than  the  mean 
value  of  u  raised  to  such  power.  This  is  readily  verified  for 
an  index  2  or  3. 

The  mean  resistance  of  a  ship  in  irregular  motion  but  at 
a  mean  speed  u  will  therefore  in  general  be  greater  than  if 
the  velocity  could  be  made  uniform  at  the  ?ame  amount. 

The  actual  modes  of  propulsion  employed  usually  involve 
slight  irregularities  in  motion,  and  these  may  be  more  or  less 
increased  by  irregularities  of  wind,  current,  tide,  depth  of 
water,  etc.  Under  usually  favorable  conditions  it  is  not 
likely  that  the  increase  of  R  due  to  this  cause  is  important, 
although  its  existence  and  explanation  are  of  interest. 


RESISTANCE. 


109 


18.    VARIATION  OF  RESISTANCE  DUE  TO  ROUGH  WATER. 

The  resistance  of  a  ship  in  a  seaway  is  known  actually  to 
be  very  considerably  greater  than  that  in  smooth  water  for 
the  same  average  speed.  The  causes  of  this  may  be  given 
under  three  heads: 

(1)  The  direct  action  of  the  waves  and  the  rolling  and 
pitching  tend  to  increase  the  irregularities  of  motion,  thereby 
affecting  the  resistance  as  described  in  the  last  section. 

(2)  The  wave-disturbance  of  the  water  tends  to  confuse 
and  disturb  the  regular  stream-line  motion,  thereby  entailing 
a  greater  drain  of  energy  from  the  ship  than  in  smooth  water 
at  equal  speed. 

(3)  The  pitching  and  rolling,   notably  the  former,   place 
the   ship   continually  in   less  favorable   positions   relative   to 
propulsion,  so  that  on  the  whole  the  mean  resistance  is  in- 
creased. 

These  causes  also  react  unfavorably  on  the  propelling 
apparatus,  lowering  its  efficiency  and  increasing  the  irregulari- 
ties of  motion.  While  this  last  result  is  secondary,  the  final 
effect  on  the  propulsion  is  the  same  as  though  the  resistance 
of  the  ship  were  virtually  increased. 

No  rules  can,  of  course,  be  given  for  the  estimate  of  such 
irregular  modifications.  It  is  a  matter  of  experience,  how- 
ever, that  the  following  qualities  are  favorable  to  uninter- 
rupted speed  in  rough  water: 

(1)  Considerable  length,  good  free-board,  and  steadiness, 
especially  absence  from  pitching. 

(2)  Large  size  and  consequently  great  weight. 

The  rea-ons  for  the  good  effects  of  the  former  are  ap- 
parent. The  good  effects  of  the  latter  are  due  to  the  increase 


110  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

of  inertia  and  the  consequent  decrease  in  the  irregularities  of 
movement. 

19.    INCREASE  OF  RESISTANCE   DUE  TO  SHALLOW  WATER 
OR  TO  THE  INFLUENCE  OF  BANKS  AND  SHOALS. 

It  is  a  well-known  fact  that  the  resistance  of  a  ship  or 
boat  at  any  given  speed  is  very  much  increased  by  shallow 
water  or  by  the  proximity  of  banks,  as  in  the  navigation  of 
canals  and  narrow  rivers.  The  primary  cause  of  this  excess 
of  resistance  is  readily  found  in  the  disturbed  stream-line 
motion.  As  a  result  of  the  decreased  cross-section  within 
which  the  stream-lines  must  be  contained,  there  arises  an 
entire  change  in  their  form  and  in  the  wave  configuration 
about  the  boat.  The  latter  in  general  is  very  much  exagger- 
ated relative  to  its  condition  in  open  deep  water,  and  the  net 
consequence  is  a  very  considerable  increase  in  the  amount  of 
energy  necessary  to  the  maintenance  of  the  configuration  of  the 
water,  and  hence  a  corresponding  increase  in  the  resistance. 

We  may  next  inquire  as  to  the  limits  within  which  these 
results  become  of  notable  significance. 

Taking  first  the  question  of  shallowness  of  water,  the  other 
dimensions  being  unlimited,  we  have  a  large  number  of 
instances  of  trial-trips  in  which  the  increase  of  resistance  due 
to  shallow  water  is  more  or  less  clearly  shown. 

White*  gives  the  following  instance: 

The  Blenheim  in  water  averaging  9  fathoms  made  20 
knots  with  15750  I.H.P.  The  wave  phenomena  were  most 
striking  and  unusual.  In  water  from  22  to  36  fathoms  with 
the  same  power  the  speed  was  21.5  knots. 

In  the  trial  of  the  U.  S.  S.  New  York  both  ways  over  a 

*  Manual  of  Naval  Architecture,  1894,  p.  467. 


RESISTANCE.  Ill 

4O-mile  course  it  was  found  that  at  the  same  point  each  way 
where  the  depth  of  water  was  charted  as  37  fathoms  the 
revolutions  fell  off  slightly,  while  the  steam-pressure  became 
increased.  The  water  over  the  remainder  of  the  course  was 
from  15  to  20  fathoms  deeper  than  at  this  point.  The  mean 
speed  was  21  knots. 

The  characteristics  of  wave-motion  in  shallow  water  have 
been  given  in  §  10.  It  is  there  shown  that  the  paths  of  the 
particles  become  elliptical  instead  of  circular,  the  ellipse 
becoming  flatter  as  the  water  becomes  shallower.  Now  it 
has  been  suggested  that  so  long  as  the  depth  is  suffi- 
cient to  allow  the  wave  corresponding  to  the  speed  of  the 
ship  to  assume  sensibly  its  trochoidal  form  and  constitution 
no  sensible  increase  in  resistance  will  result.  The  propriety 
of  considering  this  as  more  than  a  roughly  approximate  rela- 
tionship may  be  questioned,  since  we  do  not  know  that  the 
natural  wave  formed  by  the  ship  is  more  than  roughly  tro- 
choidal in  form. 

By  reference  to  §  10,  Table  V,  it  appears  that  if  the 
depth  equals  one  half  the  wave-length  the  difference  between 
the  two  kinds  of  waves  would  be  quite  negligible.  This 
leads  by  the  aid  of  §  10  (i)  to  the  following  table: 

Speed  Minimum  Depth 

in  Feet. 

10 28 

12 40 

H 55 

!f-  * 7i 

1  o 90 

20 HI 

22 135 

24 1 60 

26 188 

28 218 

30 250 


112  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

From  considerations  drawn  from  a  study  of  the  distribu- 
tion of  stream-lines  under  certain  ideal  circumstances,  D.  W. 
Taylor*  concludes  that  a  depth  of  water  equal  to  ten  times 
the  draft  is  an  outside  limit,  and  that  no  sensible  increase 
in  resistance  is  likely  to  be  met  with  so  long  as  the  depth  is 
not  less  than  six  times  the  draft. 

The  result  in  the  case  of  the  New  York,  if  reliable,  indi- 
cates that  the  influence  extends  to  a  considerably  greater 
depth  than  six  times  the  draft,  or  than  the  amount  given  by 
the  table  above.  The  point  is  one  that  cannot  be  settled 
except  by  experiment,  satisfactorily  extended  data  of  which 
are  not  yet  available. 

Turning  now  to  the  conditions  existing  in  canals,  we  have 
not  only  the  restriction  due  to  the  bottom,  but  also  that  due 
to  the  sides  as  well.  Proceeding  at  once  to  a  representative 
case,  we  may  note  briefly  the  phenomena  attending  propul- 
sion in  a  canal  of  which  the  cross-sectional  area  is  only  some 
four  or  five  times  that  of  the  greatest  section  of  the  ship. 

In  such  cases  at  moderate  speeds  the  bow  wave  is  quite 
marked,  rising  more  or  less  transversely  on  each  side  or  a 
little  ahead  of  the  bow,  and  followed  amidships  or  astern  by 
an  equally  notable  hollow.  Following  astern  is  a  train  of 
waves,  each  member  of  which  moves  with  the  velocity  of  the 
boat,  while  the  train  as  a  whole  moves  at  a  lesser  velocity,  as 
explained  in  §  12.  The  slower  the  boat  moves  the  more  are 
the  phenomena  like  those  noted  in  connection  with  motion 
on  the  open  sea.  As  the  speed  increases  the  more  definitely 
does  the  boat  place  herself  on  the  rear  edge  of  a  wave  as  in 
Fig.  43,  and  the  more  pronounced  does  the  series  of  follow- 
ing waves  become.  As  the  speed  of  the  boat  approaches 

*  Transactions  Institute  of  Naval  Architects,  vol.  xxxvi.  p.  234. 


RESISTANCE.  113 

that  corresponding  to  the  propagation  of  a  permanent  wave 
of  translation,  u  =  Vgh,  the  waves  begin  to  take  on  more 
and  more  the  characteristics  of  waves  of  translation.  They 
transmit  a  greater  and  greater  proportion  of  energy  and  the 
train  becomes  shorter  and  shorter.  The  resistance,  however, 
continually  increases  due  to  the  rapidly  growing  size  of  the 
waves  and  their  consequently  increased  demand  for  energy. 
As  the  boat  reaches  the  critical  velocity  or  passes  slightly 


beyond  it,  the  train  of  waves  contracts  to  a  single  member, 
which  takes  its  place  with  crest  amidships  or  with  the  boat 
slightly  on  its  forward  slope.  The  resistance  then  decreases, 
and  the  boat  may  be  towed  along  at  this  speed  with  a  much 
less  expenditure  of  power  than  just  before.  The  cause  of 
this  sudden  change  in  the  law  of  resistance  is  found  in  the 
relation  of  the  boat  to  the  wave.  The  necessary  energy  to 
form  the  permanent  solitary  wave  having  been  expended, 
and  the  boat  having  been  placed  in  the  most  favorable  posi- 
tion relative  to  such  wave,  the  maintenance  of  the  wave  and 
the  velocity  of  the  boat  may  be  obtained  at  a  comparatively 
small  further  expenditure.  The  resistance  for  the  most  part 
is  that  due  to  the  maintenance  of  the  wave  in  an  imperfect 
liquid,  with  the  increased  tendencies  toward  degradation  due 
to  irregularities  in  the  sides  and  bottom. 

Following  is  a  table  showing  results  of  experiments  made 
by  J.  Scott  Russell,  the  pioneer  of  experimental  investigation 
in  this  direction. 


114 


RESISTANCE  AND   PROPULSION  OF  SHIPS. 


Weight  of  Boat                 Speed  in 

Resistance 

in  Pounds.              Miles  per  Hour. 

in  Pounds. 

r  4.72 

i 

112 

5.92 

2OI 

10239         \  5.19 

275 

9.04 

250 

1  10.48 

268.5 

5.19 

250 

12579 

7.57 
8.52 

500 
400 

9.04 

280 

Speeds  at  which  these  conditions  obtain  cannot,  however, 
be  practically  maintained  in  canals,  due  chiefly  to  the  wear 
and  tear  on  the  banks  from  the  wash  of  the  wave,  and  the 
comparatively  high  speeds  necessary  for  depths  of  water 
exceeding  10  or  12  feet. 

We  are  therefore  rather  concerned  as  to  the  effect  of  the 
banks  and  bottom  on  the  resistance  at  moderate  speeds,  and 
as  to  the  necessary  ratio  between  the  section  of  the  canal  and 
that  of  the  ship  in  order  that  the  increase  in  resistance  may 
be  negligible. 

Satisfactory  data  for  a  general  discussion  of  these  points 
are  not  available.  We  have,  however,  certain  partial  results 
as  follows: 

Elnathan  Sweet*  gives  the  following  formula  as  repre- 
senting the  results  of  a  series  of  experiments  made  by  him  on 
the  Erie  Canal: 

.10303^ 
1  r-  .597  ' 

*  Transactions  American  Society  of  Civil  Engineers,  vol.  ix.  p.  99. 


RESISTANCE.  1 15 

R  —  resistance  in  pounds; 
s  =  wetted  surface ; 
v  =  speed  in  feet  per  second; 
r  =  ratio  of  section  of  canal  to  that  of  boat. 

The  speed  varied  from  2  to  3  feet  per  second,  or  from 
1.27  to  2.09  miles  per  hour,  s  varied  from  2870  to  3090; 
r  from  4.28  to  5;  depth  of  water  from  7  to  8  feet;  draft  of 
boat  from  6  to  7  feet;  displacement  from  260  to  310  tons. 

The  formula  is  a  special  case  of  the  general  formula  pro- 
posed by  J.  Scott  Russell  and  Du  Buat  for  resistance  in  a 
restricted  channel,  viz., 

R-^B' 

where  A  and  B  are  constants  to  be  determined  by  experi- 
ment. The  ranges  of  values  of  r,  vy  and  s  are  manifestly  too 
small  to  warrant  the  application  of  these  results  to  conditions 
widely  different  from  those  covered  by  the  experiments. 

Conder*  states  that  in  one  instance  where  r  =  3  a  speed 
of  7  miles  in  the  open  was  reduced  with  the  same  power  to 
5  ;  while  in  the  discussion  of  the  same  paper  Gordon  states 
that  in  the  Rangoon,  where  r  —  10,  a  speed  of  ten  miles 
was  similarly  reduced  to  7. 

M.  Camere  gives  the  following  results  cited  by  Pollard 
and  Dudebout:f 

Let  resistance  be  represented  by  a  formula 


*  Minutes  of   Proceedings   of  Institution  of  Civil   Engineers,  London 
vol.  i. xxvi.  p.  660. 

fTheorie  du  Navire,  vol.  in.  p.  458. 


n6 


RESISTANCE   AND    PROPULSION  OF  SHIPS. 


where  the  letters  have  the  same  significance  as  above  and  f  is 
a  coefficient. 


r 

Values  of  /"at  Speeds  per  Second  of 

3'.28 

4'.92 

6'.56 

I6.4 
Q.O 
3-45 

.00464 
.00836 
.0280 

.00513 

.00891 

.00564 
.00946 

More  recently  M.  de  Mas  has  conducted  extensive 
experiments  in  France,  the  results  of  which  have  been  made 
public  by  Derome.*  Among  the  many  interesting  features 
of  this  investigation  the  following  may  be  mentioned: 

Three  boats  were  tested  by  towing  on  the  Seine.  The 
ratio  of  the  section  of  the  stream  to  the  immersed  section  of 
boat  was  about  55,  for  which  the  results  should  be,  presum- 
ably, not  far  from  those  for  a  ratio  indefinitely  large.  The 
results  are  given  in  Table  I,  from  which  it  appears  that  the 

TABLE  I. 


Boat. 

Length. 

Displace- 
ment. 

Resistance  in  Pounds  at  Speeds  per  Minute  of 

98'-4 

ig6'.8 

295'.2 

393'-6 

492' 

Alma               •  •  • 

124.6 
99.4 
67.4 

286 
148 

119 
112 
112 

357 

353 
353 

783 
783 
783 

1464 
1466 
1466 

2467 
2469 
2469 

Rene    

Adrien  

resistance  was  independent  of  the  length  within  the  limits 
of  speed  and  length  employed.  The  same  three  boats 
were  afterward  tested  in  the  Bourgogne  Canal  at  speeds 

*  Proceedings  Sixth  International  Congress  on  Inland  Navigation,  1894. 
Quoted  also  by  Leslie  Robinson,  Proceedings  Institute  Mechanical  Engi- 
neers, 1897. 


RESISTANCE. 


117 


increasing  by  increments  of  49.2  feet  per  minute,  from 
49.2  to  246  feet  per  minute,  with  the  same  result  as 
regards  the  substantial  independence  of  resistance  on  length. 
The  general  truth  of  this  proposition  must  by  no  means  be 
assumed  from  these  experiments,  though  there  seems  reason 
for  believing  that,  due  to  the  peculiar  form  of  canal-boats, 
a  change  of  length  has  relatively  small  influence  on  the 
resistance. 

A, number  of  boats  with  particulars  as  in  Table  II  were 
tested  in  both  river  and  canal,  with  results  as  given  in 
Table  III. 

TABLE  II. 

PARTICULARS   OF  BOATS. 


Name. 

Length. 

Average  Breadth 
at  Midship  Section. 

Block  Coefficient 
of  Fineness. 

12^  .  3 

16.4 

.00 

Flute  

122.8 

16.5 

•  04. 

118.4. 

16.5 

•  07 

Ill  .Q 

16.1 

.Q-3 

TABLE  III. 


a 

8| 

Speed 

Speed 

-G 

98  4  Feet  per  Min. 

196  8  Feet  per  Min. 

Cross- 

*^  ri 

£'S) 

section 

*•""  o 

*r  O- 

Draft. 

Boat. 

of 

«-    5 

Resistance. 

Resistance. 

Canal. 

^  ^ 

R 

R 

A 

1) 

u 

.£'•£ 

Canal 

River 

r 

Canal 

River 

T 

^•Q 

R 

T 

R 

r 

( 

Peniche 

317.9 

86.66 

3.67 

379 

225 

1.69 

1896 

664 

2.86 

5'.  2  5  -,    Flute 

86.44 

3.68 

247 

119 

2.07 

1060 

357 

2.97 

'    Toue 

86.44 

^.68 

240 

97 

2.48 

1021 

278 

3.67 

( 

Flute 

70.30 

4-52 

154 

97 

1-59 

626 

315 

1.99 

4'.27^ 

Prussian 

68.68 

462 

119 

49 

2-45 

474 

176 

2.69 

( 

Margotat 

69.98 

4.54 

117 

46 

2.52 

434 

148 

2.94 

3'.28 

Flute 

54-04 

5.88 

1  06 

86 

1.23 

421 

284 

1.48 

n8 


RESISTANCE   AND    PROPULSION  OF  SHIPS. 


An  interesting  test  was  also  made  in  various  waterways  of 
the  boat  Jeanne,  99  feet  long,  16.4  feet  wide,  and  of  the 
Flute  class.  The  results  are  given  in  Table  IV. 

TABLE  IV. 

TEST   OF  JEANNE   IN   DIFFERENT  WATERWAYS. 


*i| 

49'.  2  per  Min. 

98'.  4  per  Min. 

i47'.6perMin- 

i96'.8perMin. 

246'  per  Min. 

Draft. 

Boat. 

Od? 

R 

R 

R 

R_ 

R 

5  S  2 

R 

r 

R 

r 

R 

y 

R 

r 

R' 

r 

„•«« 

r 

r 

•y 

•y 

*" 

Seine 

116. 

26.5 

26.5 

.00 

28.7 

28.7 

.00 

146 

146 

.00 

245 

245 

.00 

384 

384 

.00 

A 

8.31 

26.5 

.00 

28.7 

.00 

150 

•°3 

271 

.  ir 

459 

.20 

3'.  28  • 

B 

5.89 

35.3 

•33 

103.6 

.42 

227 

•56 

4o8 

.67 

692 

•83 

C 

4.64 

37-5 

.41 

108.2 

.48 

238 

•63 

452 

•85 

796 

.07 

D 

3.82 

39.7 

•50 

123.5 

.70 

284 

•95 

558 

.28 

1019 

.66 

! 

Seine 

89.3 

26.5 

.00 

83.8 

83  8 

.00 

172 

172 

.00 

295 

295 

.00 

474 

474 

.00 

A 

6-39 

3°-9 

.17 

105.8 

.26 

238 

•38 

441 

•49 

756 

.60 

4'.  27  - 

B 

4-54 

48-5 

•74 

154.3 

.84 

342 

.98 

622 

.  10 

1074 

.26 

C 
D 

3-57 
2-94 

5°-7 
7°-5 

.92 
.67 

178.6 

•13 

410 
657 

38 
.82 

814 
1389 

•75  T5O1 
4.70  2646 

" 

1:3 

| 

Seine 

72-5 

28.7 

28  7 

.00 

90.4 

90  4 

.00 

187 

187 

.00 

320 

320 

1.  00 

5" 

511 

1.  00 

A 

5-i9 

39-7 

.42 

141.1 

.56 

326 

•74 

620 

1.94 

1124 

" 

2.20 

] 

B 

3-68 

70.5 

.46 

244.7 

.71 

562 

3.00 

1047 

3-28 

1839 

u 

3-59 

I 

C 

2.90 

72.8 

•56 

264.6 

•93 

688 

3-67 

1532 

4-79 

2965 

5.8o 

While  the  information  relating  to  many  points  of  this 
problem  is  still  quite  meager,  it  is  sufficiently  well  established 
that  where  the  ratio  of  section  of  canal  to  section  of  ship  is 
not  more  than  six  or  eight,  a  very  considerable  increase  in 
the  resistance  will  result,  even  at  moderate  speeds.  Again, 
mere  sectional  area  of  canal  is  not  sufficient,  for  we  might 
have  a  very  wide  and  shallow  canal  in  which  the  ratio  of  sec- 
tions would  be  very  great,  but  in  which  we  should  have  the 
shallow-water  resistance  as  already  discussed.  In  order  to 
reduce  this  increase  of  resistance  to  a  small  or  negligible 
quantity,  it  seems  likely  that  the  transverse  dimensions  of 
the  channel  should  be  everywhere  from  six  to  ten  times  the 
corresponding  dimensions  of  the  greatest  section  of  the  boat. 

In  this  connection  it  is  interesting  to  note  that  the  influ- 


RESISTANCE. 

ence  of  fine  extremities  is  less  and  less  useful  to  reduce  resist- 
ance as  the  ratio  of  section  of  canal  to  boat  is  less.  It  is 
readily  seen  that  a  point  may  be  reached  where  the  resistance 
may  be  less  for  a  somewhat  narrow  and  full  form  than  for  an 
equal  displacement  and  equal  length,  but  with  fine  ends  and 
a  larger  midship  section,  and  hence  with  a  greater  constric- 
tion of  available  cross-section  for  stream-line  motion. 


20.  INCREASE  OF  RESISTANCE  DUE  TO  SLOPE  OF  CURRENTS. 

In  ascending  rivers  where  the  current  is  noticeable  there 
must  be  a  corresponding  up  grade  and  a  resulting  vertical 
component  to  the  movement.  The  total  energy  expended 
in  any  given  time  will  therefore  include,  in  addition  to  that 
necessary  to  overcome  the  water-resistance,  an  amount  Db, 
where  D  equals  displacement  and  b  is  the  total  change  in 
elevation  effected  during  the  interval  in  question.  The  corre- 
sponding resistance  is  naturally  D  sin  a  =  Da,  where  a  is  the 
angle  of  grade  and  always  small.  Naturally  the  amount  of 
this  part  of  the  total  resistance  in  such  a  case  is  usually 
small,  though  it  may  reach  such  an  amount  as  to  prevent  the 
ascent  of  a  river  where  the  speed  of  the  current  is  consider- 
ably less  than  the  possible  speed  of  the  ship  in  smooth  water. 

In  special  cases  OL  may  reach  a  value  of  from  .001  to 
.0015.  At  moderate  speeds  the  smooth-water  resistance 
may  not  be  more  than  .o\D  or  even  less,  so  that  it  is  readily 
seen  that  this  part  of  the  resistance  may  reach  an  amount 
considerable  in  comparison  with  that  for  such  speeds  under 
usual  conditions. 


I2O        RESISTANCE  AND  PROPULSION  OF  SHIPS. 

21.  INFLUENCE  ON  RESISTANCE  DUE  TO  CHANGES  OF  TRIM. 

In  treating  of  resistance  we  have  thus  far  considered  only 
the  horizontal  component.  It  is  quite  evident,  however,  if 
the  ship  were  to  be  maintained  in  her  statical  trim  and  draft 
and  towed  through  the  water  at  any  given  speed,  that  the1 
resultant  of  all  the  changes  of  pressure  would  in  general 
have  a  vertical  as  well  as  a  horizontal  component.  In  many 
cases,  especially  at  high  speeds,  it  becomes  important  to  take 
cognizance  of  the  existence  of  this  vertical  force.  Considered 
as  a  single  vertical  resultant,  it  usually  acts  through  a  point 
forward  of  the  statical  center  of  buoyancy.  In  the  actual 
case,  the  ship  being  free  to  yield  to  this  force,  she  will  change 
trim,  the  bow  rising  somewhat  more  than  the  stern  sinks, 
thus  decreasing  the  displacement  by  a  small  amount.  At 
certain  high  speeds  these  modifications,  especially  the  change 
of  trim,  become  very  noticeable.  At  still  higher  speeds, 
especially  with  the  cut-up  form  of  after  body  mentioned  in 
§11,  it  would  appear  that  the  trim  may  begin  to  go  back 
toward  its  normal  value,  though  the  displacement  will  prob- 
ably continue  to  decrease  with  the  increase  of  speed. 

From  another  point  of  view  we  may  consider  that  the 
motion  of  the  boat,  especially  at  high  speeds,  gives  rise  to  a 
wave  system,  and  that  the  boat  naturally  accommodates  itself 
in  trim  to  the  characteristics  of  this  system.  In  general  at 
high  speeds  such  system  will  involve  a  hollow  under  or  near 
the  stern  and  a  crest  near  the  bow,  thus  placing  the  boat  on 
the  back  slope  of  the  wave  and  naturally  giving  rise  to  the 
change  of  trim.  An  instance  is  given  by  White*  of  measure- 

*  Manual  of  Naval  Architecture,  p.  470. 


RESISTANCE.  121 

ments  taken  by  Yarrow  on  a  torpedo-boat  about  80  feet  long 
at  a  speed  of  18.5  knots.  The  change  of  trim  was  about  one- 
half  inch  per  foot  of  length  or  forty  inches  between  bow  and 
stern.  Relative  to  the  water  surface  forward  the  bow  raised 
rather  more  than  one  foot,  while  relative  to  the  surface  aft  it 
settled  less  than  six  inches.  The  remainder  of  the  change  in 
trim  is  accounted  for  by  the  change  in  the  level  of  the  water 
itself  due  to  the  wave  formation. 

Like  the  other  phenomena  attending  the  motion  of  a  solid 
in  a  liquid,  this  change  of  trim  and  of  displacement  cannot  at 
present  be  treated  by  abstract  mathematics.  The  investiga- 
tion of  the  performance  of  models  and  such  comparative 
methods  are  alone  of  use  in  aiding  to  form  an  estimate  of  the 
amount  of  such  change  in  any  given  case. 

The  influence  of  the  change  of  trim  on  resistance  is  prob- 
ably slight.  Directly  it  is  a  result  and  not  a  cause  of 
resistance.  Indirectly  it  may  be  a  cause  of  a  modification  of 
resistance  on  account  of  the  difference  which  it  may  cause  in 
the  form  of  the  immersed  body.  With  usual  forms  the  influ- 
ence of  such  slight  changes  of  this  character  as  usually  occur 
is  not  likely  to  be  important.  It  is  probable  that  the  prin- 
cipal loss  arising  from  a  change  of  trim  is  due  to  decreased 
efficiency  in  the  propulsive  apparatus  rather  than  to  actual 
increase  in  the  resistance. 

In  connection  with  these  considerations  it  may  be  pointed 
out  that  there  is  reason  to  believe  that  for  extremely  high 
speeds  relative  to  length,  such  as  are  attained  by  some 
torpedo-boats  and  fast  launches,  and  notably  by  recent 
torpedo-boats  and  torpedo-boat  destroyers,  the  coefficients  in 
the  formulae  of  resistance  which  hold  for  larger  ships  and  for 
more  moderate  speeds  undergo  considerable  change.  A 


122  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

direct  extension  of  the  results  for  lower  speeds  to  such 
extreme  values  is  not  therefore  allowable,  and  the  various 
constants  and  characteristics  of  empirical  formulae  for  resist- 
ance and  power  must  be  determined  by  direct  experiment 
under  such  conditions,  or  under  others  comparable  with 
them.  See  §  26. 

In  the  boats  mentioned  speeds  of  28  to  31  knots  have 
been  attained  on  lengths  of  from  180  to  200  feet.  The 
resistance  at  such  speeds  is  shown  to  vary  with  a  lower  index 
than  for  more  moderate  speeds,  as,  e.g.,  18  to  24  knots,  and 
the  performance  as  a  whole  seems  to  indicate  a  general 
modification  in  the  relation  of  the  resistance  to  the  attendant 
conditions. 

As  causes  of  this  change  in  relationship  we  may  look  to 
the  change  in  displacement,  and  consequently  in  wetted  sur- 
face, and  to  the  changed  location  of  the  wave  system  relative 
to  the  boat. 

Photographs  of  boats  at  very  high  speeds  show  that  under 
such  conditions  the  decrease  in  wetted  surface  may  be  quite 
considerable,  though  data  are  lacking  to  furnish  a  satisfactory 
basis  for  estimate  in  any  given  case. 

The  change  in  the  resistance  of  a  boat  in  a  canal  when  it 
approaches  a  speed  such  that  it  can  be  forced  over  on  to  the 
top  or  forward  slope  of  the  primary  wave  has  already  been 
mentioned. 

It  is  not  to  be  expected  that  these  conditions  can  be 
paralleled  in  open  water;  but  we  know  that  as  the  speed 
increases  the  location  of  the  bow  primary  wave  falls  farther 
and  farther  aft,  and  photographs  taken  under  these  conditions 
as  well  as  the  decreased  change  of  trim  seem  to  show  that  at 
a  speed  sufficiently  high  the  boat  is  forced  at  least  partly 


RESISTANCE. 


123 


over  the  primary  bow  wave,  so  that  such  wave,  once  formed, 
might  require  a  less  expenditure  for  the  maintenance  of  its 
energy  and  the  speed  of  the  boat  than  would  be  indicated 
by  an  extension  of  the  law  for  lower  speeds  to  this  extreme 
point. 

It  is  not  likely  that  this  favorable  region  in  the  law  of 
resistance  can  at  present  be  taken  advantage  of  by  other 
than  boats  of  the  types  mentioned.  For  larger  ships,  e.g.,  of 
L  =  400  feet,  such  speeds  would  exceed  40  knots,  and  the 
power  required  would  be  so  great  that  with  present  engineer- 
ing resources  its  development  on  the  weight  available  and  its 
economical  application  would  be  impracticable. 


22.  INFLUENCE  OF  BILGE-KEELS  ON  RESISTANCE. 

Bilge-keels  are  usually  located  near  the  turn  of  the  bilge, 
and  stand  normal  to  the  surface  rather  than  vertical.  They 
usually  run  for  two  thirds  or  three  quarters  the  length  of  the 
ship,  and  are  supposed  to  follow  as  nearly  as  may  be  the 
natural  stream  line  path  in  order  to  interpose  the  minimum 
resistance.  Experiments  by  Wm.  Froude  and  others  have 
shown  that  the  additional  resistance  due  to  bilge-keels  is 
slight.  Some  of  the  model  experiments  by  Mr.  Froude 
indicated  that  the  additional  resistance  due  to  the  bilge-keels 
was  less  than  that  due  to  their  surface  friction  as  usually 
computed  (§  8).  It  has  been  suggested  that  this  may  be  due 
to  the  fact  that  the  keels  are  in  a  skin  of  water  moving 
forward  more  or  less  with  the  ship,  and  that  in  consequence 
the  resistance  per  unit  area  is  less  than  that  normally  due  to 
the  relative  velocity  of  ship  and  water  as  a  whole.  A  more 
just  view  is  perhaps  to  consider  that  with  the  bilge-keels  the 


124  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

average  skin-resistance  coefficient  for  the  ship  as  a  whole  is 
somewhat  less  than  for  the  ship  without  the  keels;  so  that 
while  with  the  keels  the  total  area  is  greater,  the  coefficient 
is  less,  and  the  product  is  but  slightly  increased.  The  reason 
for  the  decrease  in  the  average  coefficient  may  be  readily  seen 
by  comparing  two  bodies  of  cylindrical  form  moving  axially 
through  the  water.  Let  one  be  smooth  and  the  other  deeply 
corrugated  or  channelled  longitudinally,  both  of  the  same 
outside  diameter.  Both  will  tend  to  set  in  motion  a  cylinder 
of  water  of  about  the  same  diameter,  and  with  the  channelled 
cylinder  this  tendency  will  be  so  effective  that  at  and  near 
the  bottom  of  the  channels  the  relative  velocity  of  the  sur- 
face and  the  water  will  be  very  slight,  and  hence  the  resist- 
ance small.  The  average  coefficient  of  surface  resistance  for 
the  channelled  cylinder  will  therefore  be  less  than  for  the 
plain,  and  while  its  area  is  much  greater,  the  total,  surface 
resistance  will  not  be  increased  in  a  proportional  degree. 
Recent  Italian  experiments  on  the  model  of  the  Sardegna* 
showed  for  speeds  of  the  ship  from  16  to  21  knots  an  increase 
of  resistance  from  I  to  5  per  cent  increasing  with  speed,  or 
with  given  power  a  decrease  in  speed  from  practically  nothing 
to  a  little  over  I  per  cent.  These  experiments  seemed  to 
indicate  a  somewhat  rapid  increase  in  the  resistance  due  to 
the  bilge-keels  from  speeds  of  about  18  knots  upward.  It 
may  be  suggested  that  this  was  due  in  large  measure  to  a 
change  of  trim  at  these  speeds  and  to  a  disturbance  of  the 
stream-line  flow,  in  virtue  of  which  the  keels  became  placed 
oblique  to  the  line  of  flow  and  thus  offered  a  certain  amount 
of  head-resistance. 

*  Rivista  Marittima,  Oct.  1895. 


RESISTANCE. 


23.  AIR-RESISTANCE. 

All  above-water  portions  of  a  ship  in  motion  are,  of 
course,  exposed  to  air-resistance.  This  from  the  irregularity 
of  the  motion  of  the  air  and  the  great  irregularity  of  the  sur- 
face  exposed  is  very  difficult  of  computation.  Except  in 
rigged  ships  with  a  high  velocity  of  the  wind,  however,  this 
effect  is  not  considered  of  great  importance,  and  usually  no 
attempt  is  made  to  estimate  its  amount.  From  data  obtained 
from  planes  moving  in  air  it  appears  that  the  resistance  may 
be  expressed  by  the  formula 


R=fAv\ 

In  this  equation  A  is  the  projected  area  of  the  surface 
exposed,  the  projection  being  made  on  a  plane  perpendicular 
to  the  direction  of  motion.  For  an  unrigged  ship  this  is 
readily  found,  being  simply  the  area  of  an  end  view  of  the 
part  of  the  ship  above  water.  Where  the  ship  is  rigged  the 
amount  to  be  taken  is  very  indefinite,  and  no  estimate  can  be 
made  except  by  comparative  methods,  for  which,  unfortu- 
nately, the  amount  of  data  available  is  insufficient.  Wm. 
Froude  was  led  to  believe  that  in  the  case  of  the  Greyhound 
the  influence  of  the  rigging  was  about  equal  to  that  of  the 
hull.  This  would  depend  very  much,  however,  on  the  nature 
and  amount  of  rigging,  and  doubtless  on  the  speed  of  the 
wind.  The  velocity  v  is  to  be  taken  as  the  longitudinal 
component  of  the  relative  velocity  of  the  wind  and  ship. 
Thus  in  calm  7-  will  be  the  speed  of  the  ship,  in  a  following 
wind  it  will  be  less,  and  in  a  head  wind  more. 

The    value    of    the    coefficient  /  rests    on    experimental 
determination.     The  experiments  of  Wm.  Froude  and  others 


126  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

indicate  that  with  pounds,  feet,  and  seconds  as  units  its  value 
may  be  taken  as  about  .0017,  and  with  pounds,  feet,  and 
knots  per  hour  as  about  .0048.  The  experimental  values  are 
derived  from  planes  of  small  or  moderate  size,  and  there  is 
considerable  uncertainty  as  to  the  values  suitable  for  use  on 
large  areas.  More  data  on  this  point  are  much  needed. 

In  the  case  of  the  Greyhound  unrigged,  with  a  relative 
speed  v  =  15  knots  the  effect  produced  was  measured  by  330 
pounds.  For  v  =  10  knots  this  would  correspond  to  about 
150  pounds.  This  is  about  1.5  per  cent  of  the  water-resist- 
ance at  the  same  speed.  With  rigging  it  was  assumed  that 
this  might  be  doubled,  so  that  in  a  calm  the  ship  would 
experience  an  air-resistance  of  perhaps  3  per  cent  of  her 
water-resistance.  For  moderate  speeds,  or  so  long  as  the 
water-resistance  varies  sensibly  as  the  square  of  the  speed, 
we  might  expect  the  relation  between  the  air  and  water 
resistance  to  remain  sensibly  constant.  For  higher  speeds 
the  water-resistance  will  increase  more  rapidly  than  the 
square  of  the  speed,  so  that  the  ratio  of  air  to  water  resist- 
ance will  decrease  at  these  speeds. 

If  instead  of  calm  air  we  suppose  a  speed  of  10  knots 
through  the  water  against  a  head  wind  of  velocity  say  30 
knots,  we  should  have  v  =  40,  and  the  air-resistance  would 
be  some  16  times  as  much  as  before,  or  unrigged  about  25 
per  cent  of  the  water-resistance  and  rigged  about  50  per  cent. 
While  these  figures  may  not  be  exact,  they  are  sufficiently  so 
for  illustrative  purposes,  and  show  plainly  the  importance 
which  wind-resistance  may  assume  in  extreme  cases. 

On  one  hand  the  constant  decrease  of  sails  and  rigging  on 
all  types  of  steam-vessels  tends  to  decrease  the  importance 
of  air-resistance.  On  the  other,  the  constant  effort  to 


RESISTANCE. 


127 


increase  speed  on  ocean-liners,  war-ships,  torpedo-boats,  etc., 
calls  increased  attention  to  all  causes  which  in  any  manner 
may  affect  the  resistance.  Additional  experiments  in  this 
direction  are  greatly  needed.  Such  experiments  might  be 
carried  out  by  allowing  a  ship  to  drift  in  a  fairly  smooth  sea 
under  the  influence  of  the  wind  alone,  and  measuring  the 
velocity  of  the  ship  relative  to  the  water  and  that  of  the 
wind  relative  to  the  ship.  Then  from  model  experiments  or 
other  known  data  relative  to  the  ship  her  resistance  at  the 
speed  attained  may  be  known,  and  this  must  equal  that  due 
to  the  air  moving  past  the  ship  with  the  relative  velocity 
observed.  Otherwise  a  ship  might  be  moored  and  subjected 
to  the  wind,  the  strain  being  measured  by  suitable  dynamo- 
metric  apparatus. 

24.  INFLUENCE  OF  FOUL  BOTTOM  ON  RESISTANCE. 

There  seems  no  reason  for  believing  that  the  condition  of 
the  bottom  of  a  ship  will  essentially  affect  anything  but  the 
skin-resistance. 

The  effect  of  foul  bottom  on  skin-resistance  has  been  dis- 
cussed in  §  9,  and  it  is  again  mentioned  here  simply  to  make 
complete  the  special  causes  which  may  affect  the  total  resist- 
ance. 


25.  SPEED  AT  WHICH  RESISTANCE  BEGINS  TO  RAPIDLY 

INCREASE. 

The  analysis  of  curves  of  resistance  usually  shows  that  for 
very  low  speeds  the  total  resistance  varies  nearly  as  the 
square  of  the  speed,  such  relation  remaining  nearly  constant 
until  a  speed  is  approached  at  which  the  resistance  begins 


128  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

rather  suddenly  to  increase  more  rapidly,  the  index  becoming: 
3  or  even  4  for  a  time,  while  at  still  higher  speeds  the  index 
may  fall  again  to  2  or  sometimes  apparently  to  even  less. 
Such  characteristics  in  the  curve  of  residual  resistance  are 
shown  in  Fig.  41.  The  speed  at  which  the  first  rapid  in- 
crease is  located  depends  chiefly,  no  doubt,  on  the  wave- 
making  features  of  the  ship;  and  while  it  is  not  expressible 
with  any  great  accuracy  by  any  simple  formula,  yet  it  is 
sometimes  considered  that  for  general  purposes  it  may  be 
taken  as  proportional  to  the  speed  of  a  wave  of  length  equal 
to  that  of  the  ship.  Hence  we  should  have 


27t 

where  b  is  some  constant.     With  knots  and  feet  as  units,  b  is 
frequently  taken  as  approximately  i,  in  which  case  we  have 

u  =  VL. 

This  must  be  considered  as  giving  simply  an  indication  of 
the  locality  of  a  region  where  the  resistance  will  begin  to 
rapidly  increase,  rather  than  the  actual  location  of  a  definite 
speed.  It  must  also  be  remembered  that  generally  the  loca- 
tion of  this  region  will  be  higher  as  the  ratio  of  length  to 
beam  is  greater,  and  as  the  prismatic  coefficient  is  less. 

26.   THE  LAW  OF  COMPARISON  OR  OF  KINEMATIC 
SIMILITUDE. 

It  is  the  purpose  of  the  law  of  comparison  to  furnish  for 
two  ships  of  similar  geometrical  form  but  of  different  size, 
and  moving  at  different  speeds,  a  relation  between  the 
residual  resistances  in  the  two  cases.  This  involves,  as  we 


RESISTANCE. 


I29 


know,  principally  the  wave-making  or  modified  stream-line 
resistance. 

This  being  the  purpose,  care  should  be  had  in  noting  the 
fundamental  assumptions  on  which  the  statement  of  the  law 
depends. 

Consider  the  stream-lines  in  an  indefinite  liquid  moving 
past  a  body  as  in  §  I.  Let  the  motion  throughout  these 
stream-lines  be  steady.  That  is,  let  there  be  no  discon- 
tinuity as  in  the  breaking  of  •  waves  at  the  surface  or  the 
formation  of  eddies  within  the  body  of  the  liquid.  The 
usual  equation  for  steady  motion  will  then  apply,  and  de- 
noting the  total  head  by  k  we  shall  have  for  any  stream- 
line, 

z  -\ [-  —  =  k.  .     A,     .      .     .     .     (i) 

2g       cr 

Now  suppose  that  we  have  a  second  body  similar  geomet- 
rically to  the  first  but  A  times  larger  in  all  directions.  The 
second  body  may  then  be  considered  as  a  magnification  of 
the  first  in  the  linear  ratio  A.  Let  it  be  placed,  like  the  first, 
in  a  liquid  of  indefinite  extent  moving  past  it  with  some 
velocity  nv,  such  that  the  stream-line  distributions  in  the  two 
cases  are  also  geometrically  similar  and  in  the  linear  ratio  A. 
It  follows  that  the  entire  systems  in  the  two  cases  are 
geometrically  similar,  and  may  be  considered  as  derived  the 
one  from  the  other  by  linear  expansion  or  diminution  in  the 
ratio  A.  For  a  stream-line  in  the  second  system  situated- 
similarly  to  that  taken  in  the  first  we  shall  have 

As  for  z, 

nv  "    vt     . 

and  we  may  put  />,   "  /, 

and         k,   "   k. 


130  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

Then  we  have 

«V       /> 

\Z  A r-  —  =   k. (2) 

2g         a 

Now  we  may  most  readily  conceive  of  a  stream-line  AC, 
Fig.  44,  as  due  to  a  head  from  some  indefinitely  large  reser- 

G 


FIG.  44. 

voir  E  situated  at  an  indefinite  distance  away.  This, 
evidently,  will  in  no  way  change  the  nature  of  things  at 
ABC.  Now  suppose  the  stream-line  continued  back  to  a 
place  F  where  the  velocity  is  inappreciable.  If  the  stream- 
line is  indefinitely  small  F  will  be  of  small  finite  area,  and  we 
may  so  consider  it  as  small  as  we  choose,  and  hence  as  very 
small  compared  with  the  dimensions  of  the  reservoir.  Then 
if  for  the  present  ABC  represent  a  stream-line  of  the  first 
system,  equation  (i)  must  hold  throughout  its  length,  and  we 
shall  have  at  F 


(3) 


In  this  equation  let  us  take  z  as  measured  from  a  base- 
line OX.  The  pressure/  will  be  simply  the  statical  pressure 
due  to  the  height  FG,  and/  -f-  a  will  in  feet  be  equal  to  this 


RESISTANCE.  13! 

height.  Hence  z  +  /  -r-  <?  is  here  equal  to  XG,  the  total 
depth  of  the  reservoir  to  the  base-line  OX.  From  (3)  this  is 
the  value  of  k.  Hence  throughout  the  entire  course  of  the 
stream-line  represented  by  (i),  the  constant  k  represents  this 
depth  GX. 

Now  the  second  system  is  a  magnification  of  the  first  in 
the  linear  ratio  A.  There  will  be  therefore  a  similar  point  F 
similarly  situated  in  a  similar  indefinite  reservoir  whose  depth 
to  a  similar  base-line  will  be  A/£,  and  this  will  be  the  value  of 
ku  the  constant  for  the  stream-line  in  the  second  system, 
whose  equation  is  given  in  (2). 

We  have  thus  far  simply  considered  the  velocity-ratio  n 
such  that  the  stream-lines  in  the  two  systems  will  be 
geometrically  similar.  We  must  now  determine  what  the 
value  of  this  ratio  will  be  in  terms  of  the  linear  ratio  A. 

Since  the  stream-lines  are  similar  throughout,  and  since 
the  flow  is  steady  and  the  condition  of  continuity  is  fulfilled, 
we  must  have  the  velocities  along  similar  lines  exactly  pro- 
portional at  similar  points.  That  is,  if  z/0,  z/,,  and  V^  V^ 
denote  velocities  in  the  first  and  second  systems  at  pairs  of 
similar  points,  we  must  have 

V       v  V        V 

^  0  ^0  V  1  ^  o 

77  —  — ,      or        -  =  — . 
v\      v\  v\        v* 

This  shows  that  n  is  a  constant,  and  that  if  we  can  determine 
the  velocity-ratio  for  any  pair  of  similar  points  the  constant 
value  of  n  will  thus  be  known.  The  velocity-ratio  at  a  pair  of 
corresponding  points  is  most  readily  found  by  aid  of  Fig.  44, 
by  considering  A  an  open  end  in  each  system.  We  shall 
then  have  p  =  o,  and  for  the  first  system 


I32  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

and  for  the  second 

»V 

2£"    "  *      ""*'" 

Hence,  dividing,  we  have 

ri*  =  A     and     #  =  4/A. 

It  follows  that  in  order  to  fulfil  the  conditions  of  simi- 
larity throughout  the  two  systems  we  must  have  a  constant 
velocity-ratio  equal  to  I/A.  Putting  this  for  n  in  (2)  we  have 

*+2^+^~  ' (4> 

whence,  comparing  (i)  and  (4),  p,  =  A/. 

The  same  relation  will  hold  for  every  other  stream-line, 
and  it  follows  that  throughout  the  entire  second  system  the 
pressure  will  be  A  times  that  at  corresponding  points  in  the 
first  system.  In  other  words,  the  magnification  in  pressure 
will  exactly  correspond  to  that  in  linear  dimension.  This 
relation  will  therefore  hold  between  the  pressures  at  corre- 
sponding points  on  the  surfaces  of  the  two  bodies,  because 
by  supposition  the  surfaces  throughout  are  in  contact  with 
the  stream-lines.  In  the  stream-lines  of  Fig.  I  the  body  is 
supposed  to  be  indefinitely  immersed,  so  that  no  surface 
changes  take  place.  All  of  the  preceding  equations  and  con- 
clusions hold,  however,  for  any  stream-line  in  a  perfect 
liquid,  and  therefore  for  a  stream-line  system  about  a  body 
partially  immersed,  so  long  as  the  conditions  of  continuity 
of  flow  are  fulfilled.  Now  it  is  due  to  the  distribution  of 
pressure  throughout  such  a  stream-line  system  that  the 
stream-line  resistance  arises.  Hence  this  resistance  will  be 


RESISTANCE.  133 

the  value  of  the  integral  of  the  longitudinal  component  of  the 
pressure  acting  at  the  inner  surfaces  of  the  stream-line  sys- 
tem. Denote  an  element  of  the  surface  by  ds  and  the  angle 
between  its  normal  and  the  longitudinal  by  6.  Then  we 
have  for  our  first  system 


=  J'p  cos  Vds. 


In  the  second  system  the  entire  distribution  of  pressure 
is  multiplied  by  \,  and  the  linear  dimensions  are  increased  in 

I  the  same  ratio.  Hence  the  ratio  between  corresponding 
elements  of  surface  will  be  Aa,  and  in  the  second  system  we 
shall  have 

R'w  =    /V  cos  0\*ds  =  V  fp  cos  6ds=  KRm. 

That  is,  with  two  systems  under  the  conditions  we  have 
assumed,  the  ratio  between  the  resistances  due  to  stream-line 
pressure  is  A3.  We  may  also  note  that  this  is  likewise  the 
relation  between  the  volumes  of  the  two  immersed  bodies, 
and  hence  between  their  displacements. 

The  law  of  comparison  thus  derived  involves  three  con- 
siderations, to  which  careful  attention  should  be  given: 

(1)  The  supposition  of  geometrical  similarity; 

(2)  The  definition  of  corresponding  speeds- 

(3)  The  statement  of  the  law. 

Geometrical  similarity  for  the  two  systems  being  assumed, 
corresponding  speeds  are  described  as  those  which  will  pro- 
duce similar  stream-line  or  similar  wave  configurations,  and 
are  defined  as  speeds  in  the  ratio  of  the  square  root  of  the 
linear-dimension  ratio  A.  Based  on  these  conditions,  we  tho* 
have  the  following  statement  of  the  law: 


134 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


A  t  corresponding  speeds  the  resistances  of  similar  ships  are 
in  the  ratio  of  the  cubes  of  like  linear  dimensions ;  or  as  the 
cube  of  the  linear-dimension  ratio  ;  or  as  the  volume -ratio. 

The  truth  of  the  law  thus  derived,  it  must  be  remem- 
bered, rests  on  the  application  of  the  formula  for  steady 
motion  to  stream-line  flow.  It  is  therefore  only  applicable 
to  that  part  of  the  total  resistance  which  is  due  to  the 
stream-line  motion,  and  hence,  so  far  as  the  above  derivation 
is  concerned,  is  not  applicable  to  skin  or  eddy  resistance.  It 
also  presupposes  the  following  as  necessary  conditions  for  its 
exact  truth: 

(1)  A  liquid  without  viscosity; 

(2)  The  absence  of  discontinuity  of  flow  such  as  would  be 
caused  by  breaking  waves  or  eddies. 

So  far  as  the  first  condition  is  concerned,  the  viscosity  of 
water  relative  to  the  velocities  with  which  we  are  concerned 
is  so  slight  that  this  departure  from  the  exact  conditions  will- 
make  no  essential  difference  in  the  application  of  the  law. 
With  regard  to  the  second  condition  we  must  distinguish 
between  the  effect  of  the  eddies,  and  that  due  to  breaking 
waves.  Eddies  will  be  formed  by  possible  irregularities  in 
the  surface,  and  as  an  expression  of  the  general  tangentia! 
action  or  skin-resistance.  Such  eddies  may,  however,  be 
considered  as  virtually  a  part  of  the  ship  so  far  as  the  trans- 
mission of  pressure  between  it  and  the  stream-line  system  is 
concerned.  The  actual  stream-line  system  involved  is  there- 
fore that  which  envelops  the  eddies.  If  the  two  ships  are 
geometrically  similar,  we  may  evidently  assume  with  safety 
that  the  existence  of  these  eddies  will  not  essentially  disturb 
the  geometrical  similarity  of  the  enveloping  systems  of  stream- 
lines. Hence  while  we  are  not  here  authorized  to  apply  the 


J?£S/S  7  'A  NCE.  1 3  5 

law  of  comparison  to  eddy  and  skin  resistance,  it  appears 
that  the  existence  of  the  eddies  as  the  expression  of  these 
two  forms  of  resistance  should  not  interfere  essentially  with 
the  application  of  the  law  to  the  systems  of  stream-lines 
actually  formed.  The  breaking  of  a  wave  involves  a  discon- 
tinuity of  stream-line  flow,  and  in  such  case  we  are  not 
strictly  entitled  to  extend  the  law  to  the  stream-line  or  wave 
resistance.  Within  the  limits  prescribed  by  this  condition, 
however,  we  may  consider  the  law  as  satisfactorily  estab- 
lished. We  may  also  consider  it  as  highly  probable  that 
even  with  breaking  waves  the  general  configurations  in  the 
two  systems  will,  at  corresponding  speeds,  still  be  essentially 
similar,  and  that  the  resistances  involved  will  still  essentially 
follow  the  same  law  of  comparison.  It  must  be  remembered, 
however,  that  while  such  may  be  the  case,  its  verification  will 
depend  on  experiment  rather  than  on  mathematical  reasoning. 

It  is  possible  to  view  the  question  of  the  law  of  compari- 
son from  an  entirely  different  standpoint,  and  thus  to  extend 
the  scope  of  its  possible  application.  It  will  be  noted  that 
the  derivation  above  given  establishes  the  law  under  certain 
limited  conditions.  It  does  not  follow,  however,  that  the 
same  law  may  not  be  applicable  under  wider  conditions.  To 
this  question  we  now  turn  our  attention. 

Assume  a  series  of  ships  all  of  similar  geometrical  form, 
but  diverse  in  dimension.  Assume  that  the  resistances  of 
such  ships  as  a  family  at  varying  speeds  may  be  represented 
by  a  general  equation  consisting  of  a  series  of  terms  all  of  the 
form 

Al^~~^vn. 
We  shall  then  have 

"«V (5) 


136  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

In  this  equation  /  is  some  typical  dimension,  necessarily 
the  same  throughout  the  series,  v  is  the  speed,  and  A  is  one 
of  a  series  of  constants  depending  on  form  characteristics, 
and  hence  constant  throughout  the  series.  Such  a  general 
equation  might  therefore  consist  of  a  series  of  terms,  as  for 
example 

R  =  Al^v  +  Bl2v2  +  CW*v*  +  Dltf,      .     .     (6) 

in  which  n  was  given  successively  the  values  I,  2,  3,  4.  It 
is  by  no  means  necessary  that  n  should  be  integral,  so  that 
terms  of  the  form  £/MzA2,  .F/'V'8,  etc.,  may  occur.  In  fact, 
so  far  as  we  are  at  present  concerned,  n  may  have  any  value 
whole  or  fractional,  and  the  number  of  terms  may  be  indefi- 
nite. 

In  regard  to  the  limits  for  the  values  of  these  exponents 
we  note  that  if  n  =  o, 


if  n  —  6, 


The  first  would  give  a  term  independent  of  speed  and 
dependent  only  on  size,  and  varying  directly  as  volume  or 
weight.  This  seems  hardly  likely,  and  we  may  consider 
n  =  o  as  the  lower  limit  of  values  of  n.  The  upper  limit 
n  =  6  gives  a  term  independent  of  dimension  and  depending 
wholly  on  speed.  As  indicated  elsewhere,  it  has  been 
thought  that  such  a  term  is  probable,  or  at  least  possible. 
In  any  event,  however,  it  is  evidently  the  upper  limit  for  n 
unless  we  admit  the  possibility  of  a  term  involving  a  decrease 


RESISTANCE. 


37 


of  resistance  with  increase  of  size,  which  seems  quite  unlikely. 
Hence  n  may  presumably  have  any  value  between  o  and  6, 
whole  or  fractional. 

Taking  therefore  (6)  simply  as  an  illustration  of  the  kind 
of  terms  to  be  found  in  such  an  equation,  we  note  ag-ain  t-hat 
A,  B,  C,  etc.,  are  form  constants,  and  remain  unchanged 
throughout  the  particular  family  of  ships.  We  assume  then 
that  it  may  be  possible  to  express  the  resistance  ©f  any  ship 
of  this  family  at  any  speed  by  substituting  in  the  appropriate 
equation  using  the  form  constants  A,  B,  C,  etc.,  and  the 
given  values  of  /and  v.  For  any  other  type  of  form  there 
will  naturally  be  another  series  of  form  constants  with  their 
series  of  values  of  exponents  for  /  and  z/,  not  necessarily  the 
same  as  in  the  equation  for  the  first  type. 

We  see  also  that  for  the  same  ship  at  varying  speeds  this 
equation  assumes  the  possibility  of  representing  the  resist- 
ance as  a  summation  of  terms  involving  the  variable  v  with 
various  exponents,  whole  or  fractional;  while  for  different 
ships  at  the  same  speed  we  should  have  the  resistance 
expressed  as  a  sum  of  terms  involving  the  variable  /  with 
another  set  of  exponents,  whole  or  fractional. 

A  little  thought  will  show  that  these  assumptions  are 
more  elastic  than  those  involved  in  the  derivation  of  the  law 
from  the  equations  of  hydrodynamics.  They  admit  a  wide 
range  of  variation,  and  an  indefinite  number  of  terms  in  the 
expression  for  R,  binding  the  exponents  of  /  and  v  only  to 
the  one  condition  that  the  exponent  of  /  plus  one  half  that 
of  v  shall  equal  3.  Taking  (5)  therefore  as  the  symbolic 
equation  let  us  compare  by  its  means  the  resistances  of  two 
ships  similar  in  form  with  a  linear-dimension  ratio  A.,  and  at 


138  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

speeds  in  the  ratio  |/A.     We  have  then,  using  subscripts   i 
and  2, 


or  , 

Hence  for  any  resistance  which  follows  the  law  as  ex- 
pressed in  (5)  the  law  of  comparison  as  above  stated  will 
hold.  Furthermore,  the  elasticity  of  equation  (5)  makes  it 
quite  possible  or  even  likely  that  it  may  properly  represent  a 
larger  portion  of  the  total  resistance  than  that  due  directly  to 
the  stream-line  or  wave-formation,  and  hence  that  the  law  of 
comparison  may  be  applicable  to  a  somewhat  greater  part  of 
the  total  resistance  than  that  to  which  the  first  mode  of 
derivation  properly  makes  it. 

Eddy-resistance,  for  example,  is  usually  considered  as 
varying  with  some  area,  and  as  the  square  of  the  speed,  §  5, 
or  as  represented  by  a  term  Z?/V.  This  is  seen  to  fall  into 
place  as  one  of  the  general  series  of  terms  in  -(5). 

Next  as  to  skin-resistance.  This  is  represented  by  a 
term  of  the  form  fAvn,  §  7,  where  f  is  a  coefficient  which 
may  decrease  with  length,  A  is  area,  and  n  is  an  exponent 
usually  less  than  2.  This  term  is  seen  to  violate  the  condi- 
tions for  members  of  (5),  and  the  amount  by  which  n  is  less 
than  2  and  the  amount  of  decrease  of  /with  increase  of  /and 
hence  of  A,  furnish  together  a  general  index  of  the  amount 
by  which  skin-»resistance  is  not  amenable  to  the  law  of  com- 
parison. With  smooth  bottom  and  the  values  derived  from 


RESISTANCE. 


139 


Froude's  experiments,  the  difference  between  the  values  of 
the  resistance  as  computed  and  as  derived  by  comparison 
may  be  considerable.  To  take  an  illustrative  case,  suppose 
/,  -j-  /,  —  4,  and  hence  v^  -r-  vl  —  2.  Also,  let  the  values  of 
/be  .01  and  .0093,  and  the  exponent  n—  1.85.  Then  the 
ratio  between  the  resistance  by  comparison  will  be  64,  while 
by  computation  it  will  be 

^16*  -93X2-*  =  53.64. 

The  ratio  of  these  is  .84,  showing  that  by  formula  the 
value  is  16  per  cent  less  than  by  comparison.  If  the  skin- 
resistance  were  about  one  half  the  total,  this  would  involve  a 
difference  of  about  8  per  cent  of  the  total  resistance. 

In  many  cases,  however,  especially  with  rough  bottom, 
the  value  of  the  exponent  is  nearly  or  quite  2,  and  the 
decrease  in  f  is  very  small.  In  such  cases  this  term  would 
likewise  fall  practically  under  the  general  law  with  the  others. 

The  question  of  the  form  of  the  terms  expressing  wave- 
resistance  has  been  discussed  in  §  14.  If  we  take  the  two 
terms  Alv*  and  Bv*,  we  find  that  each  comes  into  place  as  one 
of  the  terms  of  (5),  and  hence  both  fulfil  the  law  of  com- 
parison. 

We  will  now  give  a  series  of  equivalent  expressions  for 
the  velocity  and  resistance  ratios  for  two  similar  ships  at 
corresponding  speeds.  We  have 


__  _ 

-      =  \Tj  :  =  \A 

_  A.  i*.\'  _  /A«  fey  _  /A  y  (*,)•  _  M' 
-A\V)  -\jj  v  -\^i  w  •-  v-  •  •  • 


140  RESISTANCE  AND   PROPULSION   OF  SHIPS. 

Any  and  all  of  these,  and  others  which  may  be  written 
similarly,  are  equivalent  forms  of  the  ratios  involved  in  the 
law  of  comparison  for  resistance. 

The  actual  confidence  felt  in  the  application  of  this  law, 
either  to  the  total  resistance  or  to  the  residual  resistance, 
must  be  considered  as  resting  rather  on  the  result  of  experi- 
ment than  on  any  abstract  proof  necessarily  involving  as- 
sumptions not  perfectly  fulfilled  in  the  actual  case.  This 
agreement  may  also  be  considered  as  indicating  the  degree  of 
closeness  with  which  we  may  consider  either  the  residual  or 
total  resistance  as  capable  of  expression  by  the  sum  of  a 
series  of  terms  as  symbolized  in  (5). 

As  direct  experimental  investigation  relating  to  the  law 
of  comparison,  we  may  mention  the  experiments  referred  to 
in  greater  detail  in  §  30,  in  which  in  two  separate  cases  the 
resistances  of  a  model  and  of  its  corresponding  ship  when 
compared  according  to  this  law  were  found  to  be  in  very 
satisfactory  agreement,  the  greatest  errors  being  within  3  per 
cent. 

In  this  connection  we  may  also  note  the  experimental 
determinations  made  by  Wm.  and  R.  E.  Froude  on  the  wave- 
configurations  made  by  similar  forms  moving  at  correspond- 
ing speeds. 

The  wave  systems  due  to  a  model  of  the  Greyhound  and 
to  the  boat  herself  at  corresponding  speeds  were  compared 
by  Wm.  Froude.  Their  similarity  was  found  striking  even 
to  details  of  the  bow  wave  system.  It  was  also  found  for  the 
model  that  up  to  a  speed  of  about  1.6  feet  per  second  the 
resistance-curve  followed  almost  exactly  the  curve  of  skin- 
resistance  as  computed  by  formula.  It  was  a  matter  of 
observation  that  up  to  this  speed  the  model  moved  without 


RESISTANCE.  141 

raising  sensible  waves.  At  higher  speeds  the  total  resistance 
increased  somewhat  rapidly  over  that  due  to  skin-resistance 
alone,  and  seemed  to  follow  the  plainly  visible  augmentation 
in  the  size  of  the  wave  system.  Mr.  R.  E.  Froude  also 
mapped  with  care  the  wave-configuration  about  two  models, 
one  four  times  the  size  of  the  other,  the  larger  travelling  at 
twice  the  speed  of  the  smaller.  The  smaller  represented  a 
launch  83  feet  long  at  9  knots  per  hour,  and  the  larger  a  ship 
333  feet  long  at  18  knots  per  hour.  The  configurations  thus 
resulting  have  been  already  referred  to  and  are  shown  in 
Figs.  32  and  33,  and  their  great  similarity  in  position  and 
relation  to  the  ship  is  strikingly  seen.  In  Fig.  33  is  also 
shown  the  system  for  the  smaller  model  at  the  same  speed  as 
the  larger,  so  that  we  have  here  the  comparison  of  two  wave 
systems  made  by  different  similar  ships  at  the  same  speed. 
The  general  similarity  of  distribution  relative  to  the  water  is 
quite  apparent. 

The  first  comparison  indicates  therefore  that  similar  ships 
at  corresponding  speeds  will  produce  similar  wave-con- 
figurations relative  each  to  the  ship;  while  the  second 
indicates  that  similar  ships  at  the  same  speed  will  produce 
approximately  similar  wave-configurations  relative  to  the 
water. 

This  latter  consideration  may  lead  to  the  suggestion  that 
the  wave  pattern  for  a  given  type  of  ship  is  in  large  measure 
independent  of  the  size  of  the  ship,  and  simply  dependent  on 
character  of  form  and  on  speed.  Hence  the  energy  involved 
in  the  waves  would  be  likewise  dependent  only  on  the  same 
functions,  and  likewise  the  resistance.  It  is  these  considera- 
tions pushed  to  their  full  limit  which  gives  Bv*  as  the  form  of 
the  term  for  wave-resistance  in  §  14.  In  one  important 


142  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

feature,  however,  the  wave  pattern  cannot  be  independent  of 
the  size  of  the  ship.  This  feature  is  the  composite  stern 
system  which  in  general  depends  on  the  relative  positions  of 
the  components  due  to  the  primary  bow  and  stern  waves, 
and  this  in  turn  depends  on  the  relation  between  the  wave- 
making  length  and  the  speed.  Hence  in  part,  at  least, 
wave-making  resistance  must  depend  on  the  dimensions,  and 
its  proper  expression  may  perhaps  involve  terms  in  both  v* 
and  #",  as  already  suggested  in  §  14. 

27.  APPLICATION  OF  THE  LAW  OF  COMPARISON. 

CASE  I.  Denote  two  similar  ships  by  5,  and  5a.  If  the 
law  of  comparison  is  assumed  to  apply  to  the  entire  resist- 
ance, we  have  to  form  simple  proportions  as  expressed  by 
any  of  the  various  forms  in  §  26  (7)  and  (8).  The  given  in- 
formation must  necessarily  involve  three  terms: 

(1)  A  given  ship  5a  at,  a  given  speed; 

(2)  The  resistance  Rt  at  this  speed; 

(3)  A  similar  ship  5a  at  a  corresponding  speed. 

The  fourth  term,  R»  is  then  readily  found  from  the  pro- 
portions in  §  26  (8). 

Let  Fig.  45  be  a  graphical  representation  of  the  resist- 
ance of  a  ship  at  varying  speeds.  We  now  wish  to  show  that 
the  same  curve  may  also  be  considered  as  a  diagram  of  resist- 
ance for  all  similar  ships,  the  speed  and  resistance  scales 
being  suitably  modified. 

To  this  end  we  see  that  any  ordinate  as  AB  gives  the 

resistance  Rt  for  the  speed  v  =  OA.     Suppose  now  we  have 

a  ship  A.  times  as  large  in  linear  dimension.     Then  at  a  speed 

\v  the  resistance  will  be  W?.      Hence  if  the  scales  are  so 


RESISTANCE. 


143 


changed  that  OA  to  the  new  scale  denotes  4/;U/,  and  AB 
denotes  ICR^  the  one  will  properly  correspond  to  the  other; 
and  the  same  relation  holding  for  other  points,  we  shall  have 
to  the  new  scales  a  diagram  of  the  resistance  of  the  new  ship 


0  A  V 

FIG.  45. 

at  varying  speeds.  We  note  that  to  effect  this  change  the 
linear  amount  representing  unit  speed  must  be  divided  by  A*, 
and  the  linear  amount  representing  unit  of  resistance  must 
be  divided  by  A8.  A  curve  of  this  character,  therefore,  by 
the  proper  treatment  of  the  scales  will  represent  graphically 
for  this  type  or  family,  the  resistance  of  any  ship  at  any 
speed,  within  the  limits  determined  by  the  original  data  and 
by  the  modification  of  the  scales. 

CASE  II.  Let  OP,,  Fig.  46,  denote  the  curve  of  total 
resistance  for  a  given  ship  5,  at  varying  speeds.  Let  the 
skin-resistance  be  computed  according  to  the  methods  of  §  8. 
and  set  off  from  OPt  downward  at  the  various  speeds  as  CB 
at  A.  This  will  give  a  curve  OQ  as  the  graphical  representa- 
tion of  the  residual  resistance,  while  the  intercept  between 
OQ  and  OPt  will  show  the  value  of  the  skin-resistance.  The 


144 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


residual  resistance  as  given  by  OQ  is  then  treated  directly  by 
the  law  of  comparison,  either  by  proportion,  or  by  a  change 
of  scales,  as  just  explained.  On  the  latter  supposition  OQ 
may  also  be  considered  as  representing  to  appropriate  scales 


FIG.  46. 

the  residual  resistance  of  the  second  ship  52  at  varying 
speeds.  The  residual  resistance  of  53  being  thus  found,  the 
frictional  resistance  is  computed  as  in  §  8  by  an  appropriate 
selection  of  the  constants.  The  two  being  added  the  entire 
resistance  is  thus  determined.  For  graphically  representing 
the  entire  resistance  of  5,  we  may  proceed  as  follows:  The 
values  of  the  skin-resistance  of  5a  having  been  computed, 
find  ordinates  for  its  representation  to  a  scale  Xs  times  as 
large  as  that  used  for  St.  Lay  these  off  from  OQ,  using  a 
speed  scale  |/A  times  as  large  as  that  for  5,.  The  result  will 
be  a  curve  OP^  representing  to  these  changed  scales  the  total 
resistance  of  S,.  If  instead  of  the  ship  5,  we  have  a  model, 
the  principles  and  operations  are  exactly  similar,  and  we  may 
thus  consider  the  above  as  a  general  explanation  of  the 


RESISTANCE.  145 

method  of  connecting  the  results  of  a  model  experiment  with 
the  ship  which  it  represents. 


28.  GENERAL  REMARKS  ON  THE  THEORIES  OF  RESISTANCE. 

From  the  preceding  sections  treating  on  the  resistance  of 
ships,  it  is  quite  evident  that  pure  theory  is  quite  unable  to 
furnish  the  indications  necessary  to  the  solution  of  any  given 
problem.  The  true  function  of  theory  as  at  present  existing 
must  be  considered  as  that  of  providing  a  means  to  a  sys- 
tematic study  of  the  phenomena  attending  the  motion  of  a 
ship-formed  body  through  the  water,  and  of  establishing  the 
basis  for  an  intelligent  comparison  of  the  results  of  one  vessel 
with  those  of  another,  in  order  that  experimental  data  may 
be  made  available  for  purposes  of  prediction  and  design. 

It  may  be  here  remarked  that  the  so-called  "  form  of 
least  resistance"  in  its  abstract  sense  has  for  the  designer 
only  an  indirect  interest.  It  is  of  course  true  for  any  one 
condition  of  displacement,  character  of  surface,  state  of  the 
liquid,  and  speed,  that  there  must  be  some  form  of  least 
resistance,  or  at  least  some  form  than  which  none  can  offer 
less  resistance.  Such  form,  however,  will  doubtless  change 
with  every  change  in  the  four  fundamental  characteristics 
above,  and  with  the  question  of  resistance  must  always  be 
associated  those  of  safety,  strength,  and  carrying  capacity, 
and  often  adaptation  to  special  conditions.  The  purpose  is 
not  therefore  in  any  given  case  to  produce  a  form  of  least 
resistance  as  such,  but  rather  a  form  which  shall  the  most 
economically  combine  the  several  qualifications  in  the  par- 
ticular proportion  called  for  by  the  circumstances  of  the 
problem. 


146  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

29.  ACTUAL  FORMULA  FOR  RESISTANCE. 

Of  the  many  formulae  which  have  been  proposed  for  the 
direct  computation  of  resistance  we  shall  mention  but  few. 
In  practical  application  it  is  usually  power  rather  than  resist- 
ance which  is  computed,  though  with  the  necessary  assump- 
tions, of  course,  the  one  may  be  derived  from  the  other.  In 
Chapter  V  the  question  of  power  will  be  discussed,  and 
various  additional  formulae  and  methods  will  be  given  for  its 
determination,  from  which,  by  inverse  processes,  the  resist- 
ance might  be  determined  if  desired. 

We  have  first 

R  -  "/X3"2  (i\ 

~w    ......  w 

where  D  is  displacement  in  tons; 
v  is  speed  in  knots; 
R  is  resistance  in  tons; 
a  and  k  are  two  variable  coefficients  —  the  coefficient 

of  propulsion  and  the  admiralty  coefficient. 
These  coefficients  are  explained  in  §  46  and  §  62,  and, 
as  is  readily  seen,  the  formula  is  simply  derived  from  the 
so-called  admiralty  formula  for  power,  and  except  where  used 
with  the  law  of  comparison,  as  explained  in  §  62,  its  results 
are  only  to  be  considered  as  roughly  approximate. 

Middendorf*  has  deduced  from  numerous  experiments  on 
screw-steamers  the  following  formula,  of  which  the  first  term 
expresses  the  residual  and  the  second  term  the  skin  resist- 

ance: 

^  ..... 


Where  the  surface  is  quite  smooth,  the  exponent  of  v  in 
the  second  term  may  be  made  1.85  instead  of  2. 

*  Busley,  Die  Schiffsmaschine,  1886,  vol.  II.  p.  579. 


RESISTANCE. 


The  units  are  pounds,  feet,  and  knots.  B  is  beam;  L> 
length;  M,  area  of  midship  section ;  S,  wetted  surface;  while 
e  is  a  coefficient  to  be  taken  from  the  following  table,  using 
the  prismatic  coefficient  as  argument: 


.7o  and  under.  . 

e 

2 

.81  

e 
1.  5O 

.71 

I.QQ 

.82  

1.42 

72 

1.  08 

.83  ., 

1.32 

1.  06 

.84  , 

1.18 

.  7J. 

*  •  ;/ 

.85. 

i.  06 

I.SQ 

.86  

.OO 

.76 

.  wv^ 

1.85 

.87.. 

•  74 

.77 

1.  81 

.88  

.55 

78 

.80.. 

.70. 

1.60 

.QO.  . 

02 

.80.. 

y 

1.62 

Biisley  considers  that  this  formula  may  be  expected  to 
give  good  results  except  for  very  long,  fine  ships,  in  which 
case  the  values  are  somewhat  too  large. 

Taylor*  suggests  for  speeds  up  to  that  for  which  z/*  in 
:nots  divided  by  the  wave-making  length  is  not  more  than 
about  1.2,  the  following: 


R  = 


(3) 


rhere  the  first  term  gives  the  skin  resistance  and  the  second 
the  residual  or  wave-making.  The  units  are  pounds  for  R, 
tons  for  D,  and  feet  and  knots  for  the  other  quantities.  D  is 

*  Transactions  Society  of  Naval  Architects  and  Marine  Engineers,  vol. 
p.  143- 


148  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

displacemnt ;  L,  length;  s,  surface;  ft  the  coefficient  of  skin- 
resistance  as  discussed  in  §§  8,  9;  and  b  the  block  coefficient. 
In  this  connection  we  may  also  mention  Calvert's* 
formula  for  the  resistance  of  sections  of  propeller-blades  or 
bodies  of  similar  form  moving  at  a  slight  obliquity  to  the 
face. 

Let  v  =  speed  in  feet  per  second; 
B  =  breadth  in  feet ; 
D  =  depth  in  feet; 
6  =  angle  of  inclination; 

m  =  an  exponent  depending  on  0,  as  indicated  by  the 
following  table: 


5°.. 20 

io° 35 

20° 50 

30° 6o 

40° 69 

We  then  have 

R  = 


v  m 

50° 77 

60° 84 

70° 90 

80° 96 


90"  ............  i.  oo 


30.    EXPERIMENTAL  METHODS  OF  DETERMINING  THE 
RESISTANCE  OF  SHIP-FORMED  BODIES. 

Experiments  on  resistance  have  been  made  in  a  few  cases 
on  actual  vessels.  For  the  most  part,  however,  such  experi- 
ments have  been  limited  to  models  usually  from  8  to  15  feet 
in  length. 

Where  the  actual  vessels  have  been  used,  they  have  been 

*  Transactions  Institute  of  Naval  Architects,  vol.  xxvni.  p.  303. 


RESISTANCE. 


149 


towed  either  from  a  boom  or  off  the  quarter  of  the  towing 
ship,  in  order  to  bring  them  clear  of  the  wave  system  formed 
by  the  latter.  The  tow-rope  strain  was  then  measured  by  a 
suitable  dynamometer  and  the  speed  by  specially  constructed 
and  carefully  rated  propeller  logs.  In  this  way  tests  were 
made  by  Wm.  Froude  *  on  the  Greyhound  at  a  time  when 
tests  of  models  were  generally  considered  of  little  value. 
The  comparison  of  the  data  thus  derived  with  that  given  by 
a  model  of  the  same  ship  reduced  in  the  ratio  I  :  16  showed 
a  very  satisfactory  agreement  throughout,  and  did  much  to 
establish  confidence  in  model  experiments,  which  from  that 
time  have  been  regularly  carried  on  in  England  and  on  the 
Continent. 

A  later  comparison  of  the  resistance  of  an  actual  vessel 
-with  that  of  the  model  was  made  by  Yarrow  in  1883  f 
between  a  torpedo-boat  and  its  model.  The  range  of  speeds 
covered  by  the  comparison  was  up  to  15  knots.  Here  also 
there  was  virtual  agreement,  the  actual  values  being  some 
3  per  cent  greater  than  those  determined  by  means  of  the 
model. 

These  and  other  results,  as  well  as  the  general  agreement 
(when  special  disturbing  causes  are  eliminated)  found  between 
predictions  based  on  model  experiments  and  actual  results, 
have  all  led  to  a  high  degree  of  confidence  in  the  value  and 
reliability  of  model  experiments  for  the  determination  of  the 
resistance  to  be  expected  in  full-sized  ships. 

The  models  are  usually  of  paraffine,  shaped  on  a  special 
machine  in  which  the  block  cast  roughly  to  shape  moves 

under  rapidly-revolving  cutters  which  are  moved  by  a  panta- 



*  Naval  Science,  vol.  in.  p.  240. 

\  Transactions  Institute  of  Naval  Architects,  vol.  xxiv.  p.  in. 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 

graphic  connection  with  a  tracing-point,  the  latter  being  car- 
ried by  the  operator  along  the  water-lines  of  the  vessel  whose 
form  is  to  be  reproduced.  In  this  way  successive  water-lines 
are  traced  out  at  the  proper  successive  heights  on  the  surface 
of  the  future  model,  the  cross-sections  at  the  end  of  the 
operation  resembling  series  of  steps,  the  inner  corners  of 
which  are  on  the  surface  desired.  The  outer  corners  are  then 
worked  off  smoothly  by  hand-tools  until  the  inner  angle  of 
the  path  cut  by  the  tool  barely  remains.  The  surface  is  then 
smoothed  and  polished,  and  is  ready  for  the  trials. 

The  necessary  equipment  consists  of  a  tank  or  canal  some 
20  to  25  feet  wide,  10  or  12  feet  deep,  and  from  300  to  400 
feet  long.  A  truck  running  on  rails  spans  this  canal  and 
carries  the  necessary  apparatus  for  holding  the  model  and 
measuring  the  pull  at  the  various  speeds  attained.  The 
latter  are  measured  by  electrical  contacts  at  known  distances 
on  the  track,  and  also  through  the  revolutions  of  the  truck- 
wheels.  All  the  elements  of  the  entire  determination,  includ- 
ing seconds  of  time,  dynamometer  pulls,  changes  of  trim  and 
draft,  are  automatically  registered  on  a  drum  driven  by  con- 
nection with  the  truck-wheels.  The  models  are  taken  hold 
of  in  such  way  that  they  are  free  to  assume  whatever  draft 
and  trim  may  suit  the  conditions  from  moment  to  moment. 
If  desired  also,  special  measurements  by  photography  and 
otherwise  may  be  made  of  the  wave  systems  produced. 

For  investigating  the  effect  of  the  propeller  on  resistance 
a  special  truck  is  provided  carrying  the  propeller  connected 
with  driving-power  in  such  way  that  it  may  be  driven  at  any 
desired  number  of  revolutions  per  minute.  The  propeller 
thus  driven  and  carried  entirely  independent  of  the  model  is 
then  brought  up  into  its  proper  position  astern  of  the  model* 


RESISTANCE.  !$! 

and  the  revolutions  adjusted  until  the  total  thrust  developed 
is  equal  to  the  modified  resistance  of  the  model.  The  con- 
ditions are  then  similar  to  those  which  would  exist  were  the 

>ropeller  by  means  of  this  thrust  driving  the  model  at  the 
same  speed.  A  comparison  of  the  resistance  thus  found  with 
its  value  without  the  propeller  shows  the  increase  due  to  the 
presence  of  the  latter. 

With  the  same  apparatus  the  performance  of  model 
propellers  may  be  investigated,  and  thus  most  valuable  infor- 
mation bearing  on  the  design  of  full-sized  propellers  may  be 

letermined.     To  these  matters  we  shall  refer  later  in  §  49. 


31.  OBLIQUE  RESISTANCE. 

Ruler's    Theory.  —  Let    the    resistance    for    motion   longi- 
idinally  ahead  be  denoted  by  R^  =  A^v*,  and  that  for  motion 
transversely  by  R^  =  A^J1.     Then  if  the  actual  motion  of  a 
vessel  is  with  a  velocity  u  in  a  direction  making  an  angle   8 
with    the  longitudinal,  the  components  of  the  velocity  lon- 
gitudinally and   transversely  will  be    u   cos   6  and    u   sin   6. 
Euler  then  assumed  that  the   corresponding  components  of 
:he  total  resistance  will  be  Aj?  cos'  6  and  Aj?  sin2  0,  the 
^suiting  total  resistance  being 


R  =  u*  tfA?  cos4  6  +  A;  sin4  0,      .     . 
and  the  tangent  of  its  direction  with  the  longitudinal 

A,  sin'  0       A, 

tan  a  =  -~-     ,     =  -^  tan'  0 

A,  cos  0       A. 


(i) 


(2) 


These  equations  can  only  be  considered  as  giving  a  rough 
approximation  to  the  actual  values.     They  serve,  however, 


152 


RESISTANCE   AND    PROPULSION  OF  SHIPS. 


to  introduce  the  questions  of  principal  interest  which  are  the 
direction    and    point    of    application    of    this   total    force   R. 


FIG.  47- 

These  have  relation  to  stability  of  route  when  the  vessel  is 
moved  obliquely  as  by  a  tow-line.      In  Fig.  47,  for  a  certain 


RESISTANCE. 


153 


direction  0Fand  speed  u,  let  TS  be  the  direction  of  R,  and 
hence  ST  the  direction  of  the  applied  towing  force.  Let  M 
be  the  location  of  the  point  of  application  of  the  resistance. 
The  tow-line  must  be  attached  somewhere  in  the  line  ST. 
If  the  point  of  application  is  toward  T  from  M,  it  is  readily 
seen  that  the  couple  developed  by  any  slight  yaw  or  deviation 
from  the  given  direction  O  V  would  give  rise  to  a  moment 
tending  to  return  the  ship  to  this  direction,  and  to  the  fulfil- 
ment of  the  conditions  originally  assumed,  Under  these 
circumstances,  then,  there  will  be  stability  of  route.  If,  on  the 
other  hand,  the  tow-line  were  attached  beyond  M  toward  S, 
the  moment  due  to  a  yaw  would  tend  to  still  further  increase 
the  divergence,  and  there  would  hence  be  instability  of  route, 
with  a  tendency  on  the  part  of  the  vessel  to  swing  about  and 
go  with  the  port  side  forward.  For  each  value  of  the  direc- 
tion OV  there  will  evidently  be  a  point  M,  and  the  collection 
of  these  for  all  possible  directions  will  give  a  locus,  as  shown 
in  the  figure.  There  is  very  little  experimental  data  relating 
to  the  form  and  characteristics  of  this  locus.  Certain  general 
conclusions,  however,  may  be  reached  as  follows: 

It  is  evident  that  the  line  ST  must  touch  the  locus  at  M. 
Also,  since  physically  there  will  be  but  one  such  point  for 
any  given  position  of  OV,  it  follows  that  ST  can  touch  the 
locus  but  once,  and  hence  will  be  tangent  to  it  at  M.  Hence 
the  locus  will  be  the  envelope  of  the  series  of  lines  ST. 
From  the  results  of  Joessel  relative  to  the  location  of  the 
centre  of  resistance  for  planes,  §  6,  it  seems  reasonable  to 
assume  that  the  center  for  longitudinal  motion  will  be  at 
some  point  A,  and  that  as  the  angle  is  varied  it  will  somewhat 
rapidly  fall  aft  on  either  side,  thus  giving  a  cusp  at  this 
point.  Similarly  there  would  be  like  cusps  at  B,  the  center 


154  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

for  motion  astern,  and  at  C  and  D,  the  centers  for  motion 
nearly  transversely,  the  exact  angle  depending  on  the  amount 
of  difference  between  the  fore  and  after  bodies  of  the  ship. 
We  then  assume  the  remainder  of  the  locus  to  be  as  indi- 
cated in  the  diagram.  If  therefore  any  point  of  attachment 
on  the  ship  be  taken  for  the  tow-rope,  the  angle  between  the 
latter  and  the  keel  for  stability  of  route  will  be  determined 
by  drawing  a  tangent  from  this  point  to  the  envelope. 
There  will  be  two  or  four  such  tangents  in  general,  according 
as  the  point  is  within  or  without  the  envelope.  Observation 
will  show,  however,  which  one  is  applicable  to  any  given 
case.  This  determines  a,  the  angle  between  the  keel  and  the 
tow-rope.  In  regard  to  #,  the  inclination  of  the  direction  of 
motion,  there  seems  physical  reason  to  believe  that  the  latter 
will  always  lie  between  the  keel  and  the  direction  of  the  tow- 
rope. 

This  does  not  agree  with  the  indications  of  (2)  for  very 
small  values  of  8,  but  we  may  well  doubt  the  exactness  of 
this  equation  at  the  limit  where  one  of  the  component  veloci- 
ties is  very  small.  We  may  expect,  therefore,  that  the  rela- 
tion between  a  and  0  will  be  somewhat  as  indicated  in  Fig. 
48,  where  values  of  (a  —  0]  and  a  are  plotted  in  rectan- 
gular coordinates.  The  value  of  (a  —  0)  is  (-{-)  between  O 
and  Ay  or  for  a  approximately  between  o  and  90°,  and  (— ) 
for  a  between  A  and  B.  For  a  =  o,  180°  and  some  angle 
near  90°,  a  and  0  will  coincide  and  (a  —  6)  will  be  o,  as  indi- 
cated. If,  therefore,  the  information  expressed  by  these  two 
diagrams  is  at  hand,  any  problem  involving  the  relation 
between  the  point  of  attachment  of  a  tow-rope,  its  direction 
for  stability  of  route,  and  the  direction  of  the  motion  is 
readily  solved.  Unfortunately  we  have  no  such  accurate 


RESISTANCE. 


15* 


knowledge  for  representative  cases,  and  in  lieu  the  problem 
must  practically  be  solved  by  trial    and  error.      Guyou   has. 


VALUES  OF  ANGLE  a 

FIG.  48. 

determined  by  experiment  for  a  launch  that  the  point  A  lay- 
about  1 1  per  cent  of  the  length  forward  of  the  center. 

The  character  of  the  two  diagrams  above  will  depend  on 
the  ship  fundamentally,  and  in  the  second  place  on  the  fol- 
lowing conditions: 

(1)  Trim. — The  greater  the  trim  by  the  stern  the  greater 
the  difference  between  the  forward  and  after  bodies,  and  the 
greater  the  difference  between  the  branches  AC  and  CB  of 
Fig.  47.    The  farther  aft  as  a  whole,  relative  to  the  ship,  also, 
will  the  envelope  be  located. 

(2)  Speed. — It  is  found  by  experiment  that  when    v  in- 
creases, a  increases  slightly.      This  indicates  that  the  trans- 
verse component  of  resistance  increases  more  rapidly  than  the 
longitudinal.     The  value  of  a  is  not  therefore  independent  of 
speed,  as  in  Ruler's  equations. 

(3)  Helm. — The    position    of    the    rudder    is    capable    of 


I  $6  RESISTANCE   AND    PROPULSION  Of   SHIPS. 

changing  the  whole  character  of  the  envelope  of  Fig.  47  by 
introducing  an  additional  oblique  force  at  the  stern  and  by 
modifying  the  stream-line  motion.  It  follows  that  within 
•certain  limits  the  relative  and  absolute  values  of  6  and  a  may 
be  varied  at  pleasure  by  the  appropriate  use  of  the  helm. 

The  manoeuvring  of  ships  when  moored  by  single  cable 
or  in  a  bridle,  or  when  towed  by  single  line  or  bridle,  are 
familiar  illustrations  of  the  practical  application  of  these 
principles. 


CHAPTER     II. 
PROPULSION. 

32.  GENERAL  STATEMENT  OF  THE  PROBLEM. 


. 


THE  fundamental  problem  of  propulsion  is  to  find  a  thrust 
whereby  we  may  overcome  the  resistance  which  the  ship, 
meets  when  moving  through  the  water.  Leaving  aside  pro- 
pulsion by  sails,  the  water  about  the  ship  presents  itself  as. 
the  only  medium  by  means  of  which  this  necessary  thrust  can 
be  developed. 

Now  water  is  a  yielding  medium,  and  the  conditions  are 
entirely  different  from  those  encountered  in  pushing  a  boat  in 
shallow  water  by  means  of  a  pole,  where  a  firm  and  unyield- 
ing hold  is  obtained  on  the  bottom.  With  water  or  any  such 
fluid  medium  the  development  of  a  thrust  must  depend 
fundamentally  on  the  utilization  of  its  inertia  and  viscid 
forces.  We  may  view  the  production  of  a  thrust  from  two 
standpoints: 

(<?)  We  know  that  the  production  of  a  change  of  momen- 
tum requires  the  action  of  a  force  and  the  expenditure  of 
energy,  and  that  conversely  the  matter  acted  on  will  react  on 
the  agent  producing  the  change  of  momentum,  such  reaction 
being  in  fact  the  resistance  opposed  to  the  change  of  momen- 
tum. If  therefore  we  provide  an  agent  attached  to  the  ship 
which  shall  produce  a  change  of  momentum  in  matter  of  any 

»:ind,  such  change  of  momentum  being  directed  astern,  or  at 

157 


158  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

least  having  a  sternward  component,  there  will  result  on  the 
agent  a  reaction  having  a  forward  component,  such  reaction 
being  then  available  as  a  propulsive  thrust.  The  fundamen- 
tal conditions  necessary  are,  therefore,  that  a  change  of 
momentum  must  be  produced  in  matter  by  an  agent  attached 
to  the  ship,  and  that  such  change  must  have  a  sternward 
•component.  In  the  usual  case  water  is  the  matter  acted  on, 
and  that  in  which  the  change  of  momentum  is  produced.  A 
propulsive  thrust  would,  however,  be  given  by  a  gun  firing 
projectiles  over  the  stern  or  by  a  boy  throwing  apples  or 
stones  in  the  same  direction,  as  well  as  by  the  means  found 
more  useful  for  actual  propulsion. 

We  now  turn  to  the  second  method  of  viewing  a  thrust. 

(b)  We  have  seen  in  Chapter  I  that  no  body  can  be 
moved  through  a  liquid  without  experiencing  a  resistance. 
Let  us  then  consider  the  pair  of  bodies  made  of  the  ship  and 
a  propeller  of  some  kind,  and  let  us  attempt  to  produce  rela- 
tive motion  by  so  moving  the  latter  relative  to  the  former 
that  its  motion  shall  have  a  sternward  component.  Such 
motion  will  meet  with  a  resistance  which,  as  we  know,  is 
simply  the  resultant  of  the  distributed  system  of  pressures 
and  tangential  forces  acting  on  the  surface  of  the  moving 
body.  This  resistance  will  react  along  the  line  of  relative 
motion,  and  will  therefore  have  a  forward  component,  and 
may  hence  be  made  to  yield  a  propulsive  thrust.  It  results 
that  the  resistance  of  the  ship  and  that  of  the  propeller,  both 
taken  along  the  line  of  motion  of  the  former,  or  what  is  the 
same  thing,  along  the  line  of  motion  of  the  system  as  a 
whole,  must  be  equal.  We  shall  next  examine  geometrically 
the  conditions  attendant  on  this  second  mode  of  view. 


PROPULSION. 


159 


33.  ACTION  OF  A  PROPULSIVE  ELEMENT. 

Let  SS,  Fig.  49,  represent  the  ship  and  C  the  element, 
the  latter  being  simply  any  body  so  connected  with  the  ship 
that  its  relative  motion  is  along  a  line  AB  with  a  velocity  v. 
The  line  AB  is  not  necessarily  horizontal,  but  may  have  any 


FIG.  49. 

iirection  in  space,  and  is  simply  defined  as  making  an  angle  6 
with  the  longitudinal  HL.  As  a  result  of  this  motion  of  C 
let  the  ship  move  with  a  velocity  u  along  LH.  In  the  usual 
way  we  determine  AF  =  w  as  the  direction  and  amount  of 
motion  of  C  relative  to  the  surrounding  body  of  still  water. 
This  we  lay  off  at  CK,  remembering  that  its  direction  in 
space  will  be  determined  by  LH  and  AB.  While  the  actual 
motion  of  C  is  thus  along  CJ ,  let  its  form  be  such  that  the 
resultant  of  the  system  of  distributed  forces  acting  on  its 
surface  is  alon^  some  line  in  space  CR,  making  an  angle  y 
with  CJ.  Denote  the  inclination  of  CR  to  AB  by  //  and  to 
the  longitudinal  by  A,  and  denote  the  value  of  this  resultant 
by  R.  For  simplicity,  Fig.  49  is  drawn  as  though  all  the 
lines  were  in  the  horizontal  plane.  As  noted  above,  such  is 


l6o  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

not  necessarily  the  case,  but  with  the  angles  A,  //,  y,  and  0 
defined  as  above,  the  statement  of  the  configuration  is 
entirely  general. 

The  resistance  to  the  relative  motion  of  C  along  AB  will 
evidently  be  R  cos  jw,  and  for  the  total  work  performed  by 
the  propelling  power  we  shall  have 

W  =  R  cos  /* .  v  =  Rv  cos  // (i) 

For  the  available  thrust  we  shall  have  the  longitudinal 
component  of  R, 

T  =  R  cos  A  =  resistance  of  ship ;  (2) 

and  for  the  work  done  on  the  ship, 

Wl  =  R  cos  A .  u  =  Ru  cos  A (3) 

The  resistance  along  the  line  of  motion  of  C  relative  to 
still  water  is  R  cos  y,  and  for  the  work  thus  expended  we 
have 

W^  =  R  cos  y  .  w  =  Rw  cos  y (4) 

We  must  also  have 

W=  IV,+  W,, 

/ 

or  v  cos  fj.  =  u  cos  \~\-w  cos  y.        ...     (5). 

This  last  result  may  also  be  derived  directly  from  the  triangle 
AEF. 

Definition  of  Efficiency. — In  the  remaining  chapters  of 
this  work  we  shall  have  frequent  occasion  to  use  the  term 
efficiency.  In  its  general  sense  it  is  simply  the  ratio  of 
returns  to  total  investment;  or  of  the  useful  result  received, 
to  the  total  expense  necessary  to  its  attainment.  The  fun- 
damental definition  must  be  carefully  kept  in  mind,  for  ia 


PROPULSION. 


161 


.some  cases  it  is  not  easy  to  say  what  is  the  useful  return,  or 
what  the  total  expense. 

In  the  present  case  there  is  no  ambiguity,  and  we  have  for 
the  efficiency 

W^       u  cos  A  u  cos  A 


e  — 


W 


.     (6) 


v  cos  /*        u  cos  A  -\-  w  cos  y' 

We  may  note  that  the  three  velocties  v  cos  //,  u  cos  A, 
and  w  cos  7  are  the  projections  of  the  velocities  v,  u,  and  ze/ 
on  the  direction  of  total  resistance  CR,  and  that  the  efficiency 
is  simply  the  ratio  of  the  two  projections  of  u  and  v  along 
this  line. 

As  a  particular  case,  let  the  foregoing  equations  relate  to 
an  element  of  the  surface  of  C  instead  of  the  whole  body,  or 
otherwise  we  may  consider  the  element  as  a  small  plane. 
This  is  represented  in  Fig.  50.  In  this  diagram,  as  before,  we 


FIG.  50. 

assume  that  the  lines  are  not  necessarily  in  one  plane.     The 
total  force  R  may  here  be  represented  by  its  two  components 


162  XESISl'ANCE   AND    PROPULSION   OF  SHIPS. 

P  and  2  normal  and  tangential  to  the  plane.  Denote  the 
inclination  of  P  to  the  longitudinal  by  a  and  to  the  line  AC 
by  ft.  Then  the  inclinations  of  Q  to  the  same  lines  are 
go  -|-  a  and  90  —  /?,  respectively.  We  have  then  for  the 
thrust 

T  =  P  cos  a  —  Q  sin  a,  .      .      .      .        (7) 

and  for  the  component  of  R  along  AC 

R  cos  ,u  =  P  cos  ft  +  Q  sin  ft.  .     .     .     .       (8) 
Hence  for  the  useful  work 

Wl  —  (P  cos  a  —  Q  sin  a)&,   ....        (9) 
and  for  the  total  work 

W=  (Pcos/3+  Qsm  ft)v,  .     .     .     .     (10) 

(P  cos  of  —  Q  sin  a\u 

whence         e  ==  7-5 -5— : — -~ — : — ^r— .       .      .      .     (II) 

(P  cos  ft  -\-  Q  sin  )ff)z/ 


4-U^ 


If  Q  =  o,  we  have 

u  cos  a 


e  •=.  — 


v  cos  ft       v  cos  ft  sec  a' 


.       .        (12) 


Suppose  now  the  element  supported  on  a  smooth  unyield- 
ing surface  as  indicated  by  UV.  Then  when  it  has  moved 
relative  to  the  ship  the  same  distance  AN  as  before,  the  ship 
will  have  moved  to  A^  and  the  element  to  C^  In  such  case 
the  only  work  expended  is  that  on  the  ship.  Hence  e  must 
equal  I.  Hence  in  (12)  vt  at,  and  ft  remain  the  same,  while  «• 
must  equal  v  cos  ft  sec  a.  Or  in  other  words,  v  cos  ft  sec  of 
is  the  speed  which  the  ship  would  have  if  C  were  supported 
on  the  smooth  unyielding  surface.  This  is  readily  seen 


geom 


PROPULSION. 


I63 


etrically  if  the  lines  of  Fig.  50  are  considered  in  one 
plane;  otherwise  an  application  of  spherical  trigonometry  is 
necessary  to  a  geometrical  proof. 

The  difference  between  the  velocity  v  cos  ft  sec  a  and  the 
actual  velocity  //  is  called  the  slip.  This  we  denote  by  5 
and  the  ratio  S  -+  v  cos  /3  sec  a  by  s.  Hence  we  have 


Thisi 


and 


u  =  v  cos  ft  sec  a  —  S, 
u 


€  = 


v  cos  ft  sec  a 


=  i  —  s. 


(13) 


his  is  therefore  the  limiting  value  of  the  efficiency  when  the 
tangential  forces  Q  are  o. 

It  will  be  noted  that  the  results  thus  far  developed  are 
entirely  independent  of  any  supposition  relating  to  P  or  Q  or 
the  laws  according  to  which  they  vary ;  and  further,  that  the 
limiting  value  of  e  in  (n)  or  (12)  is  purely  a  geometrical 
function. 

Application  to  Typical  Cases. — (a)  Suppose  the  element  a 
small  plane  normal  to  the  direction  of  motion  and  moved  in 
the  direction  of  motion,  and  hence  normal  to  itself.  In  this 
case  all  tangential  forces  on  the  element  balance,  and  R  is 
simply  the  difference  of  the  two  normal  forces,  one  on  the 
front  and  the  other  on  the  back.  Referring  to  Fig.  49,  we 
have,  therefore,  o  as  the  value  of  the  angles  A,  /*,  y,  and  0. 

Hence  T  =  R, 

and  W  —  Rv  =  Tv\ 

Wl  =  Ru  =  Tu  • 

w  =  v  —  u  =  S 


e  =  -  =  1—5. 
v 


04) 


164  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

(b]  Let  the  element  be  any  solid  symmetrical  body,  or  at 
least  of  such  form  that  the  total  resistance  R  is  in  the  direc- 
tion of  motion,  and  let  the  latter  be  longitudinal.     Then  in 
Fig.  49  we  have  o  for  the  value  of  A,  yu,  7,  and  6.     Hence 
the  same  equations  apply  as  for  (a). 

(c)  Let  the  element  be  a  plane  revolving  about  a  trans- 
verse axis  attached  to  the  boat  as  in  a  radial  paddle-wheel. 
If  we  suppose  the  boat  propelled  by  a  wheel  consisting  of  a 
large   number  of  such  elements,  we  know   that  in  order   to 
obtain  the  necessary  reaction  the  elements  must  move  back- 
ward somewhat  relative  to  the  water,  and  that  if  u  be  the 
speed  of  the  boat  and  v  the  peripheral  speed  of  the  paddles, 
v  will  be  greater  than  u.     If  v  were  equal  to  u,  it  is  readily 
seen  that  the  element  relative  to  the  water  would  describe  a 
cycloid,  the  motion  being  in  fact  exactly  as  though  the  boat 
were    carried  on    wheels  of  a  radius    equal    to    that    of    the 
element,   such  wheels    rolling  without  slip  on  a    rigid  plane 
horizontal  foundation.     In  the  actual  case  the  motion  will  be 
as  though  the  wheels  were  reduced  in  diameter  in  the  ratio 
of  u   to   v>    and   likewise    rolled    without    slipping    on    such 
reduced  diameter.     As  a  result  the  element  will  now  travel 
in  a  curtate  trochoid,  as  illustrated  in  Fig.  51.      In  this  figure 
the  paddle  is  denoted  by  the  heavy  line,  and  the  lower  por- 
tion of  the  path  of  a  point  P  is  shown  by  the  curve  PQRS. 
The  line  of  centers  is  at  AB,  and  the  slip  being  taken  at  20 
per  cent,  the  effective  radius  OO  is  80  per  cent  of  OR.     The 
rolling  circle  will  then  touch  the  line  CD,   which  therefore 
will   contain   the   instantaneous  centers  of  the   path   of  the 
point  P.     The  location  of  the  center  of  the  wheel  for  succes- 
sive angular  positions  of  the  element  being  as  shown  by  the 
numbers  along  ABy  the  corresponding  instantaneous  centers 


PROPULSION. 


I65 


8 


1 66  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

are  located  at  points  vertically  underneath  and  similarly 
numbered  on  the  line  CD.  The  location  of  these  centers 
serves  to  show  in  any  position  the  direction  of  motion  of  the 
point  P  relative  to  still  water,  and  hence  approximately  the 
like  motion  of  the  entire  blade.  Taking  the  blade  as  a 
whole,  the  various  elements  lying  between  the  inner  and 
outer  edges  will  have  varying  values  of  r  and  hence  of  v,  and 
hence  will  travel  in  different  trochoidal  curves,  all  deter- 
mined, however,  by  the  conditions  and  in  the  manner  above 
described. 

For  this  case  in  Fig.  50  we  should  have 

*  =  8; 

J3  =  o. 
Hence  in  (7)  we  have 

T  =  Pcos  0—  Q  sin  6.      .  *.      .      .     (15) 

The  sign  of  sin  6  changes  as  the  element  passes  through 
the  vertical.  Since,  however,  before  reaching  the  vertical 
the  element  is  inclined  on  one  side,  and  after  passing  it  the 
inclination  is  on  the  other  side,  it  follows  that  Q  itself  also 
changes  sign,  and  hence  that  the  horizontal  component  of  Q 
is  always  subtractive.  Hence  the  value  of  T  will  always  be 
correctly  given  by  (15),  considering  6  as  always  plus.  We 
have  also  in  (8) 

R  cos  fi  =  P. 

Hence  W,  =  (P  cos  0  -  Q  sin  ff)u ; 

W=  Pv\ 

(P  cos  6  —  Q  sin  0)u 
*='  -~ • 


PROPULSION. 


I67 


(d)  Let  the  element  be  a  plane  moving  as  on  the  end  of 
an   oar,  but  suppose  the  plane  of  the  oar  to  remain   always 
vertical.      This  case  is  readily  seen  to  be  similar  to  the  radial 
paddle-wheel,  and  so  far  as  the  general  equations  go,  those 
given  above  in  (c)  apply  here  as  well. 

(e)  Let  the  element  be  a  plane  always  vertical  and  revolv- 
ing about  an  axis  fixed  to  the  boat,  as  in  certain  forms  of 
feathering  paddle-wheels   (§  38).     The  path  of  the  element 
relative  to  the  water  will  be  the  same  as  before,  but  we  shall 

have 

a  =  o; 

fi=e. 

Hence  in  equations  (;)-(ii) 
T=  P\ 

R  cos  p  =  P  cos  0  +  Q  sin  0  ; 
W,  =  Pu\ 
W—  (P  cos  6+  Q  sin  ff)v\ 

_  Pu  _ 

"  ' 


(/)  Let  the  element  be  as  in  (c)  and  (e)t  but  so  connected 
that    its    plane   shall    always    pass  c 

through  the  upper  point  of  the 
•circle  which  it  traverses  relative  to 
the  boat,  as  in  certain  forms  of 
feathering  paddle-wheel.  (See  Fig. 
52.)  In  this  case  we  have 


or  =  - 


ft**-. 


FIG.  52. 


1 68  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

Hence  in  equations  (7)-(n) 

6  6 

r=/>cos-- 0sin-; 

e  e 

R  cos  )W  =  P  cos  -  +  Q  sin  - ; 

2  2 

W,  —  Tu,      W  =  Rv  cos  yw; 

^Pcos  -  —  Q  sin  -Ju 
and  ^ 


Let  us  now  compare  the  limiting  efficiencies  of  these 
three  styles  of  paddle-wheel  elements,  assuming  that  u  and  ^ 
are  the  same  in  each  case.  The  latter  condition  is  necessary 
in  order  to  make  a  comparison  possible,  and  is  perfectly 
allowable  and  natural,  since  it  merely  requires  an  adjustment 
of  the  amount  of  surface  to  the  size  or  resistance  of  the  ship. 

Remembering  that  our  equations  apply  simply  at  a  given 
instant,  we  have, 

u  cos  0 
forfc),  e=— — ; 

for  0),  e  = 


V  COS 


for  (/), 


The  order  of  excellence,  so  far  as  efficiency  goes,  is  seen  to 
be  (e)y  (/),  (c).  For  (/)  the  efficiency  is  constant  or  inde- 
pendent of  6,  and  therefore  this  is  the  value  for  all  parts  of 
the  revolution  during  which  the  element  is  acting,  and  there- 


PROPULSION.  109 

fore  the  value  for  this  element  as  a  whole.  For  other  like 
elements  the  value  will  vary  inversely  as  v,  and  hence 
inversely  as  their  radial  distances  from  the  axis  of  revolution. 
The  resultant  limiting  efficiency  is,  of  course,  a  mean  of 
these  elementary  values. 

For  (c)  and  (e),  however,  the  case  is  different.  The 
efficiency  for  an  entire  stroke  will  be  the  useful  work  divided 
by  the  gross  work  for  the  same  period.  Hence  we  should 
have, 

2Pu  cos  B 

for  (c)j  e  = 

for  (e)t  e  = 


cos  0 

Now  P  being  really  a  function  of  0  and  the  velocities,  it 
is  evident  that  no  simpler  general  expression  can  be  obtained 
except  by  making  special  assumptions  as  to  the  form  of  P. 
It  seems  reasonable  to  believe,  however,  that  under  similar 
conditions  the  order  of  excellence  with  regard  to  both  limit- 
ing and  actual  efficiency  might  be  expected  to  remain  the 
same  as  above. 

34.  DEFINITIONS  RELATING  TO  SCREW  PROPELLERS. 

If  one  line  revolves  about  another  perpendicular  to  it  as 
an  axis,  and  at  the  same  time  move  its  point  of  contact  along 
this  axis,  then  points  in  this  revolving  line  will  describe 
helical  curves,  and  the  line  as  a  whole  or  any  part  of  it  will 
generate  a  helical  surface. 

A  screw  propeller  may  for  our  present  purpose  be  defined 
as  consisting  of  one  or  more  blades  having  on  the  rear  or 
driving  side  an  approximately  helical  surface,  such  blades 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 

being  joined  to  a  common  boss  or  central  portion  through 
which  they  receive  their  motion  of  rotation  in  a  transverse 
plane  relative  to  the  ship. 

'Fig.  53  shows  a  four-bladed  right-hand  propeller. 


FIG.  53- 

A  propeller  is  said  to  be  right-hand  or  left-hand  according 
as  it  turns  with  or  against  the  hands  of  a  watch  when  viewed 
from  aft  and  driving  the  ship  ahead. 

The/tf^  or  driving-face  of  a  blade  is  the  rear  face.  It  is 
that  face  which  acts  on  the  water,  and  which  in  return 
receives  the  excess  of  pressure  which  gives  the  driving  thrust. 

The  back  of  a  blade  is  on  the  forward  side.  Care  must 
be  taken  to  avoid  confusion  in  the  use  of  these  terms. 

The  leading  and  folloiving  edges  of  a  blade  are  respectively 
the  forward  and  after  edges. 


PROPULSION.  171 

The  diameter  of  a  propeller  is  the  diameter  of  the  circle 
swept  by  the  tips  of  the  blades. 

The  pitch  is  the  axial  distance  between  two  successive 
convolutions  of  the  helical  surface.  In  an  actual  propeller 
the  term  should  be  understood  as  strictly  referring  to  a  small 
clement  of  the  driving-face  only.  With  this  understanding 
the  pitch  may  be  defined  as  the  longitudinal  distance  which 
the  ship  would  be  driven  were  such  element  to  work  on  a 
smooth  unyielding  surface,  as  for  example  the  corresponding 
helical  surface  of  a  fixed  nut.  Pitch  is  thus  seen  to  be 
purely  a  geometrical  function  of  the  propeller.  Its  value 
may  vary  from  point  to  point  over  the  entire  driving-face,  or 
it  may  be  constant.  In  the  latter  case  the  propeller  is  said 
to  be  of  uniform  pitch.  If  it  increases  as  we  go  from  the  hub 
toward  the  outer  circumference,  the  pitch  is  said  to  increase 
radially.  If  it  is  greater  on  the  following  than  on  the  leading 
edge,  the  pitch  is  said  to  increase  axially.  The  latter  mode 
of  variation  is  usually  implied  by  the  simple  term  increasing 
or  expanding  pitch. 

The  area,  developed  area,  or  helicoidal  area  of  a  blade  is 
the  actual  surface  of  the  driving-face.  For  the  propeller  it 
is,  of  course,  the  sum  of  the  areas  of  the  blades. 

The  projected  area  is,  correspondingly,  the  area  of  the 
projection  on  a  transverse  plane  of  one  blade  or  of  all  the 
blades. 

The  disk  area  is  the  area  of  the  circle  swept  by  the  tips  of 
the  blades. 

The  boss  or  hub  is  the  central  body  to  which  the  blades 
are  all  united,  and  which  in  turn  is  attached  to  the  shaft. 


172 


RESISTANCE  AND   PROPULSION   OF  SHIPS. 


35.    PROPULSIVE  ACTION   OF  THE  ELEMENT  OF  A  SCREW 

PROPELLER. 

It  follows  from  the  definition  in  the  preceding  section 
that  a  helicoidal  path  must  lie  in  the  surface  of  a  cylinder,  and 
that  the  pitch  is  equal  to  the  distance  between  two  successive 
convolutions  of  the  helix  measured  along  the  same  element 
of  the  cylinder.  It  is  shown  in  geometry  that  if  the  surface 
of  such  a  cylinder  is  developed,  the  helical  path  becomes  a 
straight  line,  the  diagonal  of  the  rectangle  representing  the 
portion  of  the  cylinder  containing  one  convolution.  Thus  in 
Fig.  54,  if  the  cylindrical  surface  be  cut  along  the  line  AB, 
and  the  lower  part  rolled  out  and  flattened,  the  helix  AHB 
will  become  the  diagonal  AC  of  a  rectangle  ABCD,  Fig.  55. 
In  this  rectangle  AB  denotes  the  pitch  and  AD  the  circum- 
ference of  the  circle  of  diameter  AD,  Fig.  54.  The  angle 
CAD,  Fig.  55,  is  called  the  pitch-angle,  and  the  ratio  AB  -~ 
AD,  Fig.  54,  is  called  \.}\z  pitch-ratio. 

A  radius  AL  moving  so  as  to  always  rest  on  LL  and 
AHB,  Fig.  54,  at  the  same  time  keeping  perpendicular  to 


FIG.  54- 


FIG.  55- 


LL,  will  then  describe  a  helical  surface  such  as  we  are  now 
concerned  with.      Let  us  consider  a  small  element  of  this  sur- 


PROPULSION. 


173 


face  at  the  radius  LA.  If  this  element  acted  without  slip,  as 
already  defined,  then  it  is  evident  that  for  one  revolution 
the  ship  would  be  driven  ahead  a  distance  AB  and  the 
element  would  travel  in  the  helical  path  AHB.  In  the 
actual  case  due  to  the  yielding  of  the  water  the  longitudinal 
component  of  the  motion  will  be  ABl  instead  of  AB,  while 
the  circular  component  will  remain  the  same.  The  element 
will  hence  actually  traverse  a  helical  path  AH1B1  instead  of 
AHB.  Let  this  be  represented  in  the  developed  diagram, 
Fig.  55>  bv  AE.  Then  DE=AB^,  the  amount  of  longi- 
tudinal motion,  and  CE  =  BBlt  the  amount  of  slip.  The 
angle  CAE  is  often  termed  the  slip  angle. 

Let  diameter  AD,  Fig.  54,  =  d\ 

radius  AL  "  "     =  r\ 

pitch  AB  "  "     =/; 

pitch-ratio  =  c\ 

revolutions  =  N\ 

area  of  element  =  dA  ; 

angle  CAE,  Fig.  55,      =  0. 

Then,  using  tke  nomenclature  of  Fig.  50,  we  have 

0  =  90° ; 
CAD=  a; 
PAD  =  p. 
Hence         a  +  ft  =  90° ; 


and 


tan  ex  = 


p  =  2nr  tan  a\ 

P        P 

c  =  -7  =  —  •=.  n  tan  a. 
d       ir 


174 


RESISTANCE   AND    PROPULSION  OF  SHIPS. 


We  will  now  apply  the  general  equations  of  §  33  to  the 
element  under  consideration. 
We  have  for  the  thrust 

T  =  P  cos  a  —  Q  sin  a (i) 

Also,  R  cos  ft  =  P  sin  a  +  Q  cos  a ; (2) 

W,  =  Tu  =  (P  cos  a  —  Q  sin  a)u ;    .      .      .      .  (3) 

W  =  Rv  cos  }A  =  (P  sin  a  -|-  Q  cos  «)z/ ;     .      .  (4) 

(P  cos  a  —  Q  sin  «)« 


(/*  sin  a-\-  Q  cos  «)«> 
For  the  limiting  value  when  Q  =  o,  as  in  previous  cases, 
u  cos  a  u 


(5) 


e  = 


v  sin  a      v  tan 


We  have  next  to  derive  expressions  for  the  forces  P  and 
Q  considered  as  determined  by  the  motion  of  the  element. 

We  may  first  note  that  these  forces  will  vary  directly  with 
the  density  of  the  liquid.  Inasmuch,  however,  as  this  factor 
enters  equally  into  the  resistance  of  the  propeller  and  that  of 
the  ship,  it  is  evident  that  so  far  as  the  relations  of  the  two 
are  concerned,  the  design  of  the  propeller  will  be  independent 
of  the  density.  That  is,  the  same  propeller  should  drive  the 
ship  with  equal  efficiency  in  either  fresh  or  salt  water.  In 
any  event  the  density  factor  will  be  provided  for  by  the  con- 
stants used  to  express  the  values  of  these  forces. 

The  actual  direction  of  motion  of  the  element  relative  to 
still  water  is  determined  by  the  angles  a  and  0. 

Now  let  Fig.  55  represent  to  an  appropriate  scale  the  dis- 
tances moved  per  minute  instead  of  per  revolution. 


PROPULSION. 


Then 


AD  =  2nrN  —  v\ 

DC  =  pN  =  2  nrN  tan  a ; 

DE=pN(\  —  s)  =  27rrNtan  a  (i  —  s)  =  u\ 


AC—  27trNVi       tana  a. 


Taking  §5(1)  and  §  6  (5)  as  the  general  expression  of  P> 
we  should  have 


P=a  dAAE   sin  0. 


EK 


But     sin  0  =  -j-        and     .ZsA"  =  CE  cos  a  =  jr/7V  cos  a. 


Hence      P=adAAEEK. 

Whence,  substituting  and  reducing,  we  find 


sin  a 


i  —  ^a  tana  a  adA. 


Likewise  from  §  7,  taking  the  relative  gliding  velocity  of 
water  and  element  as  AC,  and  assuming  a  variation  as  the 
square  of  the  speed,  we  have 


Q=f~AC*dA  = 


+  tan3  atfdA. 


The  coefficient  /  will  depend  on  the  length  of  the  ele- 
ment, character  of  surface,  and  angle  0. 

Now  let  dT,  dU,  and  dW  denote  for  this  element  the 
thrust,  the  useful,  and  the  total  work,  instead  of  T,  Wiy 
and  W. 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 

Then  from  (i),  (3),  (4),  we  have 


=  $i?r*N*dA(as  sin  a  cos  a  Vi  +  (i  -  -  s)*  tan2  a 

—  /sin  a(i  -\-  tan2  <*)"); 


dU=  87rVW3(l  -  JX/4(<W  sin2  a  '/i  +  (i  -  j)1  tan3  a 

—  f  sin  <*  tan  <*  ( i  +  tan2  a 


=  S7i3r*NadA(as  sin2  «  Vi  +  (i  -  ^)a  .tan3  a 

+  /  cos  a  ( i  +  tan8  a)  ). 


Put  sin3  aVi  +  (i  —  sf  tan3  a  =  B\ 

sin  a  tan  a  (i  -j-  tan3  a)  =  C\ 
cos  a  (i  -f-  tan3  a)  =  E. 
Whence  C  =  E  tan3  a. 

Then  the  above  equations  become 

dT  =  47r*r*N*dA(as£  cot  a  -  fC  cot  *) ;  .      .     .      (6) 
dU  =  8arVW(i  -  s)dA(asB  -  fC)\       ....      (7) 

Whence 

; 

0 

asB-fC_,         .7s 


Now  put  2r  =  yp  where  y  is  a  variable  denoting  the  loca- 
tion of  the  element  in  terms  of  the  pitch.      It  is  called  the 


PROPULSION. 


177 


diameter  ratio,  and  is  evidently  the  reciprocal  of  the  pitch- 
ratio  c  for  the  element  in  question.  For  the  propeller  as  a 
whole  the  values  of  c  and  y  will  be,  of  course,  those  of  the 
outer  element.  Where  we  wish  to  especially  distinguish, 
these  limiting  or  outer  values  we  shall  denote  them  by  c^ 
and  jjv 

We  then  have 


dT  —  n*p*N*y*fdA\^sB  cot  a  —  C  cot  a)  ; 


-  C)  ;    .      . 


(10) 
(i  i) 
(12) 


36.  PROPULSIVE  ACTION  OF  THE  ENTIRE  PROPELLER. 

If  the  expressions  in  the  preceding  section  for  dT,  dU, 
and  dW  could  be  integrated  over  the  surface  of  the  blade,  we 
should  have  values  for  the  entire  T,  U,  and  W.  Assuming 
p  constant,  these  may  be  represented  by 


T  =  *p*N*y*(asB  cot  a  -  fC  cot  a]dA  ;        .      .     (i) 
U  =  n*p*N\i  -  s}fy\asB-/C)dA  ;      ....     (2) 

(3) 
Also,  dividing  (2)  by  (3),  we  have,  similar  to  §  35,  (9), 

/  £sB  -  C}dA 


.     .     .     (4) 


1/3  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

Of  the  variables  involved  in  this  integration  we  may 
remark  that  y,  s,  B,  C,  E,  and  dA  are  determined  by  the 
geometry  of  the  propeller  and  by  the  slip.  They  are  there- 
fore readily  known.  It  is  not  the  same,  however,  with  a 
and/.  While  these  are  called  constants,  there  is  no  reason 
for  assuming  that  their  values  will  be  the  same  for  different 
•elements  of  the  surface,  or  for  different  aggregations  of  such 
elements.  In  fact  such  indications  as  we  have  point  toward 
a  decided  variation  in  value  with  different  values  of  the  total 
area,  pitch-ratio,  slip,  thickness  of  blade,  condition  of  sur- 
face, and  location  of  .element. 

In  regard  to  the  coefficient  /it  must  be  remembered  that 
it  here  represents  not  merely  a  surface  effect,  but  rather  the 
resultant  force  in  the  direction  of  the  plane  of  the  driving- 
face.  Due  to  the  increase  of  thickness  from  the  tips  inward, 
we  should  expect  a  corresponding  increase  in  the  value  of  this 
coefficient. 

The  coefficient  for  direct  resistance  is  taken  in  the  form  a 
sin  0,  and  the  supposition  of  constancy  in  the  value  of  a 
throughout  the  integration  virtually  assumes  that  the  normal 
resistance  varies  directly  with  the  surface.  This,  however, 
can  only  be  even  approximately  true  up  to  a  certain  point. 
We  have  specified  in  §  32  the  two  general  methods  of  con- 
sidering the  propulsive  action  of  an  element.  Thus  far  we 
have  made  use  of  the  second  one  only.  In  considering  the 
variation  in  value  of  this  coefficient  a,  we  shall  find  it  useful 
to  here  briefly  consider  the  propulsive  action  of  the  element 
of  the  screw  propeller  viewed  from  the  first  standpoint.  In 
§  37  and  §  38  a  more  general  discussion  of  the  application  of 
this  method  will  be  found. 

From   this  point  of  view  the  action   of  the  element  is  to 


PROPULSION,  179 

Accelerate  in  a  sternward  direction  the  water  acted  on.  This, 
for  the  element,  is  a  cylindrical  shell  of  water  of  radius  equal 
to  that  of  the  location  of  the  element,  and  of  thickness 
determined  by  its  height.  For  the  propeller  it  would  be  a 
cylindrical  body  of  water  of  diameter  sensibly  equal  to  that 
of  the  propeller,  and  with  a  hollow  core  of  diameter  sensibly 
equal  to  that  of  the  hub.  As  we  shall  show  later,  the  maximum 
amount  of  acceleration  depends  on  the  slip  and  on  the 
geometrical  configuration  of  the  element  or  of  the  propeller. 
If,  now,  we  consider  the  element  very  narrow,  as  for  example 
a  part  of  a  narrow  lath-like  blade,  the  amount  of  acceleration 
produced  either  by  the  element  or  by  the  blade  as  a  whole 
will  be  very  much  less  than  this  geometrical  maximum  above 
referred  to.  The  thrust  obtained  will  therefore  be  corre- 
spondingly less.  As  we  increase  the  width  of  such  a  blade 
the  factor  a  will  at  first  be  nearly  constant  for  each  element 
of  increase,  so  that  for  a  time  the  thrust  will  increase  directly 
with  the  area.  At  length,  however,  as  we  approach  a  condi-j 
tion  where  the  maximum  acceleration  and  maximum  thrust 
are  nearly  attained,  the  increase  in  thrust  for  each  additional 
element  of  area  will  be  less  and  less,  or  in  other  words,  the 
thrust  will  increase  at  a  slower  and  slower  rate  relative  to  the 
increase  of  area.  We  shall  therefore  finally  reach  a  width  of 
blade  such  that  no  additional  increase  will  add  sensibly  to  the 
thrust.  It  is  evident  that  as  we  thus  increase  the  area  from 
our  very  narrow  initial  blade  the  average  value  of  a  or  the 
thrust  received  per  unit  area  will  steadily  decrease,  at  first 
slowly  and  then  more  rapidly.  It  even  seems  possible  that, 
under  certain  conditions,  after  having  passed  a  certain  point 
the  increase  of  area  might  be  attended  by  an  actual  decrease 
in  total  thrust  due  to  tin:  increasing  confusion  of  the  streams 


180  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

of  water  acted  on.  These  points  will  be  again  referred  to  in 
§  49.  We  wish  now  simply  to  indicate  that  this  coefficient 
will  probably  vary  with  the  total  area  in  the  general  manner 
above  described. 

Again,  the  actual  values  of  4>  due  to  the  influence  of  the 
thickness  of  the  leading  edge  may  perhaps  be  quite  different 
from  the  values  determined  geometrically,  and  the  law  of 
variation  with  the  sine  of  the  angle  is,  again,  not  quite  exact. 

It  is  therefore  evident  that  the  coefficient  a  instead  of 
being  constant  will  probably  vary  for  different  locations  and 
configurations  of  the  element,  for  different  values  of  the  slip, 
and  for  different  amounts  of  total  area  of  blade.  If  therefore 
we  wished  to  make  direct  use  of  these  expressions  for  T,  U, 
and  J^as  they  stand,  we  should  need  a  series  of  experimental 
values  of  these  coefficients  appropriate  to  the  circumstances 
of  the  case  in  hand.  There  is,  however,  no  such  set  of  values 
available,  so  that,  even  if  it  were  desirable,  no  direct  use  can 
at  present  be  made  of  these  expressions  in  the  form  given 
above. 

It  is  not  without  interest,  however,  to  note  in  passing 
the  results  of  a  special  examination  into  the  question  of  the 
distribution  of  the  values  of  a  and /radially  along  the  blade. 
While  no  direct  experimental  determination  of  the  values 
for  an  element  are  available,  we  have  in  Froude's  experimen- 
tal investigation  on  propellers*  the  integrated  results  corre- 
sponding to  T,  Uj  W,  and  e.  As  we  shall  refer  more 
especially  to  these  experiments  at  a  later  point,  it  is  sufficient 
at  present  to  state  that  the  data  thus  obtained  give  the 
values  of  T,  U,  and  e  for  a  series  of  propellers  of  constant 

*  Transactions  Institute  of  Naval  Architects,  vol.  xxvu.  p.  250. 


PROPULSION.  181 


diameter  and  variable  pitch  (and  hence  of  variable  value  j,) 
at  a  series  of  slips.  Now  it  is  evident,  taking  any  one  slip, 
that  some  distribution  of  the  values  of  a  and  /substituted  in 
the  fundamental  formulae  (i),  (2),  (4)  will  give  the  observed 
results,  and  hence  that  by  a  process  of  trial  and  error  an 
approximation  to  the  probable  distribution  may  be  reached. 

Referring  also  to  (4),  it  is  seen  that  e  depends  solely  on 
a  -i-  /,  and  hence  that  the  distribution  of  this  ratio  may  be 
determined  inversely  from  the  values  of  e  for  various  external 
or  outer  values  of  y.  Likewise  from  (i)  it  is  seen  that  T 
may  be  expressed  as  a  function  of  /"and  a  -T-/,  and  hence,  if 
the  latter  is  first  determined,  the  distribution  of  the  values  of 
/  may  be  inferred  from  that  of  the  values  of  T.  The  values 
of  a  -T-/  and  /being  thus  found,  the  values  of  a  immediately 
follow.  Without  entering  into  the  details  of  the  operation 
we  may  state  the  results  as  follows: 

Taking  the  slip  as  20  per  cent,  it  was  found  that  a  distri- 
bution of  a  -7-  /as  given  by  AB,  Fig.  56,  would  give  by  sub- 
stitution in  (4)  the  observed  values  of  the  efficiency.  Next, 
that  a  distribution  of  /as  shown  in  CD  would,  in  connection 
with  the  values  of  a  -tr  f,  give  by  substitution  in  (i)  the 
observed  values  of  T.  The  resulting  values  of  a  are  shown 
by  EF.  It  was  also  clearly  indicated  that  no  other  distribu- 
tion widely  differing  from  this  in  character  could  fulfil  the 
given  conditions.  The  values  of  /vary,  as  we  should  expect; 
while  those  of  a  present  a  more  complex  variation,  due 
probably  to  the  varying  width  of  blade  at  different  radial 
distances  on  the  same  propeller,  and  at  points  having  the 
same  value  of  the  diameter  ratio  on  different  propellers,  and 
to  other  causes  which  do  not  appear  from  the  data  at  hand. 

For  most  purposes,   however,  we  are  not  so  much  con- 


1 82  RESISTANCE   AND    PROPULSION   OF  SHIPS. 


15 


13 


12 


10 


ioo/ 


IOO.T 


.3  .4  .5  .6 

DIAMETER  RATIO 

FIG.  56. 


1.0 


PROPULSION. 


cerned  with  the  distribution  of  the  values  of  a  and /as  with 
mean  values,  or  rather  with  values  of  the  integrals  of  (i),  (2), 
(3),  (4).  These  integrals  depend  on  the  values  of  yl  and  s, 
and  on  the  shape,  proportions,  and  area  of  the  blade.  Sup- 
posing for  the  present  the  latter  elements  constant,  it  is 
evident  that  if  we  can  find  the  values  of  these  integrals  for  a 
sufficient  number  of  values  of  y^  and  s,  we  shall  be  able  to 
obtain  by  interpolation  the  thrust,  useful  work,  and  efficiency 
corresponding  to  any  given  set  of  conditions. 

We  will  now  briefly  refer  to  the  experiments  most  avail- 
able for  this  purpose.  In  the  paper  above  referred  to- 
Mr.  Froude  gives  the  results  of  a  series  of  experiments 
which  he  had  carried  on  at  the  Admiralty  Experimental  Tank 
for  the  purpose  of  investigating  the  relation  of  thrust  and 
efficiency  to  pitch-ratio  and  slip.  The  propellers  were  of 
2,  3,  and  4  blades,  and  were  all  of  8.16  inches  diameter,  and 
of  pitch-ratios  1.225,  J-4>  x'8,  2.2.  The 
blades  were  elliptical  in  developed  form,  as 
shown  in  Fig.  57.  The  hub  took  the  place 
of  the  inner  part  of  the  ellipse,  and  the  form 
was  slightly  changed  at  this  point  to  give  a 
somewhat  wider  line  of  attachment  between 
hub  and  blade.  The  maximum  width  of  the 
blade  was  .4  the  radius.  These  propellers 
were  tested  at  various  values  of  the  slip.  It 
was  found  that  within  the  limits  of  the  experiment  the  results 
might  be  expressed  quite  closely  by  the  following  laws: 

(a)  For  constant  slip  and  varying  values  of  yl  the  thrust 
may  be  expressed  by  an  equation  of  the  form 


1 84  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

(b)  For  constant  efficiency  and  varying  values  of  yl  the 
thrust  may  be  expressed  in  the  form 

T  =  By*. 

These  equations  were  made  use  of  to  reduce  the  observa- 
tions to  order  and  to  eliminate  minor  instrumental  and  obser- 
vational errors,  and  also  to  interpolate  for  intermediate 
values  of  yl  and  s.  Where  the  nature  of  the  law  seemed  to 
authorize  it,  the  values  were  extended  slightly  beyond  the 
limits  of  the  experimental  determination. 

The  data  thus  determined  were  then  put  into  graphical 
form  in  sets  of  curves,  one  of  which  gave  T  as  a  function  of 
the  slip  for  varying  values  of  the  pitch-ratio  <:,.  These  curves 
were  so  plotted  as  to  be  independent  of  absolute  dimension, 
so  that  by  the  necessary  assumptions  as  discussed  in  §  49  and 
by  the  introduction  of  the  proper  functions  of  any  proposed 
dimensions  they  become  applicable  to  any  given  case. 

In  regard  to  the  curves  of  efficiency,  it  was  found  that  by 
expressing  e  as  a  function  of  s  for  varying  values  of  j/,  or  <;,  a 
series  of  curves  were  obtained  which  were  essentially  the 
same  curve  plotted  to  varying  horizontal  scales.  It  therefore 
became  possible,  by  an  appropriate  choice  of  horizontal  scale 
for  each  pitch-ratio,  to  bring  all  of  these  efficiency  curves  into 
essential  identity,  and  hence  to  use  one  for  all.  This  fusion 
of  the  various  efficiency  curves  was  thus  obtained  by  substi- 
tuting for  the  abscissa  s  a  quantity  called  abscissa  value  such 
that 

Abscissa  value  =  sf(c^). 

Any  values  of  s  and  cl  being  then  given,  and  the  corre- 
sponding form  of  /"(O  Demg  known,  the  abscissa  value 
follows,  and  thence  from  the  curve  the  efficiency  is  known. 


PROPULSION. 


I35 


As  the  subject  is  to  be  treated  somewhat  differently  here, 
we  shall  refer  the  reader  to  the  original  paper  for  further 
details,  especially  as  to  the  methods  given  for  applying  these 
results  to  the  problem  of  design. 

Mention  may  also  be  made  of  a  series  of  experiments 
on  screw  propellers  by  Thornycroft  and  Barnaby.*  The 
general  results  of  these  experiments  also  closely  correspond 
with  those  by  Mr.  Froude,  and  as  the  latter  are  the  more 
extensive  we  shall  use  them  as  the  source  for  our  experimen- 
tal values. 

In  order  to  place  the  equations  (i),  (2),  (3),  (4)  in  the 
most  convenient  form  for  application  to  these  experiments  we 
proceed  as  follows: 

Let  b  denote  the  varying  width  of  blade,  and  bl  the  maxi- 
mum width;  also  r,  as  before,  the  radius  of  any  element  and 
rl  the  outer  radius.  Then 


r    _2r    _py  _ 

=  ~   =  c'y' 


or 


r  =• 


The  propellers  had  four  blades.      Hence  we  have 

dA  =  4fbdr  =  ^bc^r^dy  =  2bdcldy. 
For  an  elliptical  blade  as  described 


=  ±l'Vr1r-r"  =  2j14'V1-y-*iy.        •     •     (5) 


Hence        dA  =  4&,<tc,  Vc,y  —  c'y*  dy. 


(6) 


*  Transactions  Institute  Civil  Engineers,  vol.  en,  p.  74. 


1 86 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


This  relates  the  area  of  the  elliptical  blades  to  the 
diameter  of  the  propeller  and  to  the  maximum  width  b^  We 
wish  next  to  substitute  for  this  some  form  of  relationship  in 
which  the  geometrical  character  of  the  contour  of  the  blade 
will  not  be  explicitly  involved.  To  this  end  we  relate  the 
area  of  the  blades  directly  to  nd*  -~  4,  or  to  the  disk  area  of 
the  propeller.  To  this  end  put 


A—h\ — -1     or     /z  =  ^-r,. 


(7) 


We  shall  call  h  the  area-ratio. 

It  is  found  that  the  area  of  a  blade  such  as  represented  in 
Fig.  57  is  about  .891  the  area  of  the  ellipse.  Hence  for  one 
such  blade  we  have 

Area  —  .SQITT -1  =  .35^,. 

4  2 

Hence  for  the  entire  propeller  in  this  case 

A  — 


or 


nhd* 
db,  =  .714^  =  -714—^—  =  - 


(8) 


Substituting  this  in  (6)  we  have,  as  the  more  general  relation 
desired, 

(9) 


dA  —  2. 


^y  —  c?y*  dy 


In  the  present   case  we  remember  that  £,  =  .4^  =  .2d. 
Hence  from  (8)  we  have 


hd*  =  1.78^=  .SS^1     or     A  =  .356. 


PROPULSION. 


1 87 


We  must  bear  this  in  mind  as  the  standard  value  of  h  for 
the  experimental  propellers. 

Substituting  for  dA  in  (i),  (2),  (3),  we  have 


cot  a  —  fC  cot  ai)Cl  Vc,y  —  c*y*dy\\ 


y\asB  -  fC)c, 


*f  dy\\ 


7i 3  2 . 2  4/1  /*y  ( a  sB 


t  Vc~y  -  ctf  dy\. 


The  quantities  within  the  brackets  are  seen  to  be  func- 
tions of  the  proportions  of  the  propeller  and  of  the  slip,  or 
explicitly  of  y  and  s.  Denote  for  any  given  case  their 
values  by  //,  K,  L,  respectively.  We  must  here  remember 
that  we  are  not  at  present  applying  these  equations  to  pro- 
pellers in  general,  but  simply  to  the  propellers  of  this  series 
of  experiments,  in  which  the  size,  proportion,  and  shape  of 
the  blades  remained  constant,  and  yl  and  s  were  the  only 
variables.  Hence  we  may  properly  consider  that  the  values 
of  H,  K,  and  L  are  functions  of  these  variables  only.  We 
have  therefore 

*H;     .....     (10) 


K 
e=L' 


00 

(12) 

(13) 


So  far  as  any  ultimate  use  is  to  be  made  of  these  equa- 
tions, we  may  simply  consider  U  and  e.  W,  if  desired,  is 
readily  found  by  dividing  U  by  e,  and  T  by  dividing  U  by 


the  speed  u. 


1  88  RESISTANCE   AND    PROPULSION   OF  SHIPS, 

We  are  therefore  to  derive  actual  values  of  K  by  substi- 
tuting in  equation  (i  i)  the  values  found  from  Froude's 
experiments.  To  avoid  the  presence  of  large  numbers  in  our 
results  we  first  substitute  for/,  d,  and  N,  as  follows: 

,  _  pitch  in  feet         / 

- 


_ 


o  m 

diameter  in  feet  _     d 
10  "  fo' 


,  _  revolutions  per  min.  _     N 
10  ~  10' 

Then  let  us  denote  the  resulting  value  of  K  by  the  two 
factors  kl,  where  k  is  a  function  of  the  slip  s  and  is  intended 
to  represent  the  influence  of  this  variable,  and  /  is  a  function 
of  pitch  or  diameter  ratio,  c  or^1,,  and  is  intended  to  represent 
the  influence  of  this  variable. 

We  have  then 

U=(p'N')*d"kl.     .     .     .     .     .     (14) 

Now  putting  into  (14)  an  actual  value  of  U  as  given  by 
multiplying  Froude's  curves  of  thrust  by  the  speed  «,  and 
reducing  to  horse-power  units,  we  are  able  to  derive  the  value 
of  the  combined  factor  kl  corresponding  to  the  particular 
values  of  yl  and  s  assumed.  If  this  be  done  for  a  large  num- 
ber of  values  of  y^  and  s,  we  shall  have  a  corresponding  series 
of  values  of  kl.  These  we  may  naturally  seek  to  express, 
each  as  a  function  of  the  variable  on  which  it  depends. 

Keeping  in  mind  proposition  (a)  above,  it  is  evident  that 
we  may  put 

/=(/,-•  V)  .......     05) 


PROPULSION. 


,89 


An  examination  of  the  series  of  values  of  kl  then  showed 
that  the  values  of  k  could  be  very  nearly  expressed  by  the 
quadratic  formula 

.      .      .     (16) 

Tabular  values  of  k  and  /  will  be  found  in  §  50.  In  the 
form  of  an  equation  the  final  result  becomes 


805*  -  .6Bs')(y,  -  .17). 


In  this  equation  U  is  in  horse-power  units;  /',  N't  and  d 
are  as  defined  above;  while  s  and  yl  are  to  be  expressed  as 
decimals. 

The  resulting  values  of  £/will  then  be  found  to  closely 
represent  all  values  falling  within  the  limits  of  s  and  yl 
covered  by  the  experiments. 

The  formula  given  in  (17)  is  therefore  simply  empirical, 
expressing  the  results  of  the  experiments  noted  above,  and 
therefore  fundamentally  applicable  simply  to  the  propellers 
there  employed.  In  Chapter  IV  we  shall  consider  the  exten- 
sion of  this  equation  to  propellers  in  general. 

We  now  turn  to  the  question  of  efficiency.  From  (4)  it 
appears  that  in  general  e  is.  a  function  of  s,  a,  f,  B,  C,  E,  or 
of  a,  f,  s,  yr  If  we  now  assume  that  the  influence  of  the 
general  variation  of  a  and  f  on  e  is  unimportant,  it  would 
follow  that  we  might  express  in  general  e  as  a  function  of  j, 
or  c,  and  s, — the  former  as  fixing  the  character  of  the  pro- 
peller, the  latter  the  nature  of  its  use.  This  assumption  is 
undoubtedly  inadmissible  when  the  character  of  the  blade  as 
regards  shape  and  area-ratio  h  widely  varies,  and  the  influ- 
ence of  dimension  alone  is  perhaps  not  unimportant.  As  a 


190  RESISTANCE  AND    PROPULSION   OF  SHIPS. 


working  basis,  however,  let  us  first  assume  the  blades  all 
similar  and  of  the  standard  form  and  proportions  as  used  in 
Froude's  experiments,  and  let  us  neglect  the  influence  due  to 
actual  dimension.  Then,  knowing  the  efficiency  of  the  model 
propellers,  that  of  any  other  similar  propeller  will  be  the  same 
as  that  for  the  model  of  the  same  pitch-ratio  and  working  at 
the  same  slip.  If  next  we  assume  the  area-ratio  h  and  the 
shape  of  blade  to  vary,  the  results  determined  will  be  corre- 
spondingly less  reliable.  Still  from  such  indications  as  we 
have  it  seems  likely  that  for  slight  variations  in  area-ratio  or 
shape  the  influence  on  the  efficiency  would  be  small.  In 
default  of  more  exact  information,  therefore,  we  shall  take 
the  results  of  these  experiments  as  our  standard  of  efficiency, 
remembering  that  as  we  depart  from  the  shape  and  propor- 
tions of  the  blades  there  used  the  results  will  correspondingly 
decrease  in  probable  accuracy. 

As  above  mentioned,  the  efficiencies  of  the  model  pro- 
pellers were  expressed  by  Froude  as  a  function  of  c  and  s  by 
means  of  the  "  abscissa  value,"  through  the  agency  of  which 
the  efficiency  curves  for  all  values  of  c  were  brought  into 
coincidence.  For  our  present  purposes,  however,  we  prefer 
to  put  the  information  in  a  somewhat  different  form,  as 
follows : 

Determining  by  Froude's  diagrams  the  values  of  c  and  s 
for  a  series  of  constant  values  of  e,  it  was  found  on  plotting 
the  results  that  each  locus  of  c  and  s  for  constant  e  was  very 
nearly  a  straight  line,  and  that  the  bundle  of  lines  radiate 
very  nearly  from  a  single  point.  The  actual  character  of  a 
locus  of  c  and  s  thus  determined  is  fixed  by  propositions  (a) 
and  (£),  which  of  course  are  approximate,  and  not  physically 
exact.  Starting  from  these  propositions,  it  may  be  shown 


„ 

;n 


PROPULSION.  191 

tl:at  the  actual  locus  is  not  a  straight  line,  and  that  its 
equation  is  quite  complex.  Between  the  limits  of  the  experi- 
ments, however,  the  equation  represents  a  locus  which  is  very 
nearly  rectilinear.  The  character  thus  found  by  geometrical 
determination  is  therefore  justified  by  the  nature  of  the 
propositions  on  which  the  fundamental  diagrams  are  based. 
The  same  result  is  also  implicitly  involved  in  Froude's 
determination  of  the  fact  that  the  efficiency  curves  for  vary- 
ing pitch-ratios  may  by  means  of  the  abscissa  value,  as 
referred  to  above,  be  reduced  all  to  the  same  form,  and  hence 
one  made  to  serve  for  all. 

The  diagram  deduced  as  above  explained  is  shown  in 
Fig.  58.  The  approximate  focus  of  the  lines  is  at  about 
c  =  —  6.8,  s  =  —  .21,  so  that  the  approximate  equation  to 
the  set  of  lines  is  s  +  -2  I  =  n(c  +  6.8).  The  actual  lines  of 
the  diagram  are  those  which  most  closely  represent  the  points 
determined  from  the  original  data,  and  do  not  therefore 
exactly  coincide  with  a  bundle  emanating  from  the  focus 
mentioned.  The  difference  is  very  slight,  however,  and  is 
presumably  much  less  than  the  necessary  difference  between 
the  experimental  and  true  values  in  any  given  case,  and  is 
therefore  quite  insignificant.  This  diagram  may  be  carefully 
examined  with  profit.  Geometrically  it  may  be  regarded  as 
a  series  of  level  lines  cutting  the  surface  which  would  repre- 
sent e  as  a  function  of  c  and  s.  A  plane  perpendicular  to  the 
axis  of  s  as  indicated  by  AB  would  cut  out  a  curve  of 
efficiency  for  constant  slip  as  a  function  of  pitch-ratio,  while 
a  plane  perpendicular  to  the  axis  of  c  would  cut  out  a  curve 
of  efficiency  for  constant  pitch-ratio  as  a  function  of  slip. 
S.ich  curves  are  shown  in  Figs.  59  and  60.  It  is  the  latter 
which  is  made  use  of  by  Mr.  Froude.  Looking  again  at  the 


I92  RESISTANCE   AND    PROPULSION   OF  SHIPS. 


diagram  in  general,  it  is  seen  that  it  readily  indicates  the 
relation  to  e  of  any  propeller  of  standard  form,  given  its  pitch- 
ratio  and  slip;  and,  vice  versa,  it  shows  the  range  of  values  of 

.63  .64  .65 


31 


-30 


.20 


67 


18 
16 
.14 

.12 

.10 
1.0      1.1      1.2      1.3      1.4      1.5      1.6      1.7      1.8      1.9      2.0      2.1      2.2      2.3      2.4      2.5 

PITCH   RATIO 

FIG.  58.—  DIAGRAM  OF  EFFICIENCY  AS  A  FUNCTION  OF  SLIP  AND  PITCH 

RATIO. 

these  variables  within  which  any  particular  efficiency  may  be 
expected.  Thus  the  line  for  69  per  cent  shows  the  relative 
values  of  c  and  s  along  which  the  best  efficiency  was  found, 
while  the  area  between  the  two  lines  for  68  per  cent  shows 
a  region  or  range  of  values  of  c  and  s  within  which  the 
efficiency  was  not  below  68  per  cent.  Again,  it  shows  plainly 


PROPULSION. 


193 


the  more  rapid  falling  off  as  we  approach  small  slip  rather 
than  large,  and  in  consequence  that  it  is  better  to  work  in  a 
region  of  slip  greater  than  that  for  maximum  efficiency 
rather  than  less.  Further  uses  of  the  diagram  will  appear 
with  application  to  problems  of  design,  but  the  above  will 


,u 


.U7 


.04 


.61 


.00 


Slip  25* 
Slip  30* 
Slip  20* 


Slip  15* 


1.3 


1.4 


2.2 


1.6  1.8  2.0 

PITCH  RATIO 

FIG.  59.— CURVES  OF  EFFICIENCY. 


2.4 


suffice  to  indicate  the  main  lines  along  which  its  study  may 
be  directed. 

The  reader  should  perhaps  at  this  point  be  cautioned  that 
the  absolute  values  of  the  efficiency  here  given  are  probably 
less  accurate  than  the  relative  values.  Unavoidable  errors 


194 


RESISTANCE  AND   PROPULSION   OF  SHIPS. 


enter  into  the  measurement  of  many  of  the  quantities  in- 
volved in  efficiency,  and  in  addition  some  modifications  were 
made  by  Mr.  Froude  in  order  to  reduce  the  values  to  agree- 


( 


5 


\ 


'  AON3IOUJ3 


w 


ment  with  proposition  (&)  above.  The  numbers  here  given 
should  therefore  be  considered  not  as  absolute  values,  but  as 
close  approximations;  while  the  indications  as  to  change  of 


PROPULSION, 


'95 


value  for  change  of  condition  may  be  accepted  as  presumably 
somewhat  more  accurate  in  character. 


37.  ACTION  OF  A  SCREW  PROPELLER  VIEWED  FROM  THE 
STANDPOINT  OF  THE  WATER  ACTED  ON  AND  THE 
ACCELERATION  IMPARTED. 

In  the  treatment  of  the  screw  propeller  in  §§  35,  36, 
we  have  preferred  the  method  there  followed  to  that  which 
involves  a  consideration  of  the  amount  of  water  acted  on  and 
its  acceleration,  because  the  former  seems  more  fruitful  prac- 
tically, and  to  furnish  a  more  valuable  means  of  relating 
theory  to  practice  than  the  latter.  The  latter  method,  how- 
ever, has  chiefly  attracted  the  attention  of  mathematicians 
and  theoretical  investigators,  and  it  may  be  well  to  consider 
briefly  some  of  the  leading  features  of  the  theory  from  this 
point  of  view,  especially  as  they  are  of  general  interest  as 
well. 

The  water  acted  on  must  necessarily  travel  in  helical 
paths.  That  is,  its  velocity  will  have  a  longitudinal  and  a 
rotary  component.  The  longitudinal  component  is  that 
which  directly  yields  the  reaction  from  which  comes  the 
thrust.  The  rotary  component  as  well  as  the  longitudinal 
absorbs  work  from  the  propelling  agent.  Its  existence  also 
gives  rise  to  centrifugal  force,  due  to  which  the  pressure  is 
less  in  the  interior  of  the  column  than  at  its  outer  boundaries. 
The  water  in  coming  aft  into  the  zone  of  influence  of  the 
propeller  gradually  has  its  velocity  increased,  and  a  certain 
amount  of  acceleration  is  therefore  received  before  the  water 
reaches  the  propeller  itself.  Mr.  R.  E.  Froude  *  has  shown 

*  Transactions  Institute  Naval  Architects,  vol.  xxx.  p.  390. 


196  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

in  the  limiting  case  where  no  rotation  is  produced,  and  where 
the  propeller  blades  are  so  narrow  that  little  or  no  sensible 
acceleration  can  be  received  in  the  time  taken  to  pass  through 
the  propeller  itself,  that  one  half  the  acceleration  will  be 
received  forward  of  the  propeller  and  one  half  in  the  race  aft 
of  the  propeller. 

With  rotation,  however,  we  shall  have  in  and  aft  of  the 
propeller  the  defect  of  pressure  in  the  interior  of  the  column 
due  to  centrifugal  force.  In  the  actual  case,  therefore, 
somewhat  more  than  one  half  of  the  acceleration  will  be  pro- 
duced forward  of  the  propeller,  a  certain  amount  while  pass- 
ing through,  and  the  remainder  aft  of  the  propeller. 

It  is  also  evident  that  the  higher  the  pitch-ratio  the  more 
the  rotation,  and  vice  versa.  Hence  with  propellers  of  high 
pitch-ratio  we  may  expect  that  most  of  the  acceleration  will 
be  received  forward  of  the  propeller,  while  with  low  values 
of  pitch-ratio  the  proportion  there  received  may  be  but  little 
more  than  one  half  the  total.  The  column  as  a  whole  must, 
for  continuity  of  flow,  decrease  in  cross-section  as  it  passes 
through  the  propeller,  and  to  such  distance  aft  as  the  velocity 
increases. 

If  the  amount  of  water  acted  on  could  be  known  and  the 
total  acceleration  produced,  the  thrust  could  be  immediately 
determined,  and  knowing  the  speed,  the  useful  work  would 
follow.  The  total  work  would  be  the  sum  of  the  useful,  plus 
the  work  represented  by  the  kinetic  energy  imparted  to  the 
water.  The  latter  representing  the  waste  is  seen  to  depend 
on  the  amount  of  water  acted  on  and  on  the  initial  and  final 
velocities.  It  is  also  clear  that  we  may  divide  the  energy 
spent  in  the  wake  into  three  parts,  as  follows:  (i)  that  due 
to  longitudinal  change  of  velocity;  (2)  that  due  to  circular 


PROPULSION. 


I97 


change  of  velocity  or  rotation;  (3)  that  due  to  turbulence  and 
eddies.  The  portion  (i)  is  of  course  necessary  to  the  produc- 
tion of  a  useful  thrust,  while  (2)  and  (3),  though  not  funda- 
mentally necessary,  cannot  be  entirety  avoided.  The  energy 
involved  in  the  race  rotation  has  been  made  the  subject  of 
examination  by  R.  E.  Froude,*  who  shows  by  certain  ideal 
limiting  cases  that  its  influence  on  efficiency  will  be  practically 
negligible  except  with  large  values  of  the  slip  and  high  pitch- 
ratios. 

If  in  a  simple  ideal  case  M  denotes  the  amount  of  water 
acted  on  and  v  the  longitudinal  change  of  velocity,  §  42,  (4), 
then  the  thrust  will  be  proportional  to  Mv  and  the  energy 
involved  to  MV*.  Now  for  a  given  value  of  Mv  it  is  readily 
seen  that  Mv1  will  be  a  minimum  when  M  is  a  maximum  and 
v  a  minimum.  That  is,  for  a  given  thrust  the  energy  due  to 
longitudinal  acceleration  is  least  when  the  mass  acted  on  is 
large  and  the  acceleration  is  small.  For  high  efficiency,  there- 
fore, this  consideration  points  to  the  use  of  large  propellers 
acting  at  small  slip  rather  than  the  reverse.  This  corresponds 
in  §  35  to  the  result  consequent  on  the  assumption  of  the 
decrease  or  absence  of  tangential  forces.  Actually  the  exist- 
ence of  race  rotation,  eddies  and  turbulence  due  to  skin-resist- 
ance, etc.,  as  well  as  structural  considerations,  limit  the  size 
and  slip  for  the  best  efficiency  as  already  shown  in  §  36. 
Compare  also  §  49. 

It  is  thus  possible  to  discuss  the  action  of  the  screw  pro- 
peller from  the  standpoint  of  the  present  section.  In  the 
actual  case,  however,  none  of  the  terms  here  involved  can  be 
exactly  related  to  known  quantities,  so  that,  as  in  the  other 
mode  of  treatment,  we  are  thrown  upon  experiment  for  the 

*  Transactions  Institute  of  Naval  Architects,  vol.  xxxm.  p.  265. 


IQ  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

ultimate  control  of  whatever  formulae  may  be  proposed.  We 
give,  however,  in  §  38,  as  an  illustration,  the  treatment  of 
the  paddle-wheel  from  this  point  of  view.* 

38.  THE  PADDLE-WHEEL  TREATED  FROM  THE  STAND- 
POINT OF  §  37. 

The  paddle-wheel  may  be  investigated  in  the  same 
general  way  as  the  screw  propeller  in  §§  35,  36.  The 
lesser  relative  importance  of  this  instrument  of  propulsion 
does  not,  however,  seem  to  justify  the  necessary  work,  and 
furthermore  we  have  no  such  experimental  data  for  the  final 
control  of  our  equations  as  with  the  screw  propeller.  We 
will  therefore  consider  it  briefly  from  the  standpoint  men- 
tioned in  §  32  and  referred  to  more  especially  in  §  37. 

The  thrust  is  measured  by  the  sternward  momentum  of 
the  water  imparted  by  means  of  the  paddle. 

Let  ^0  be  the  initial  mean  velocity  of  the  water  in  f.  s. ; 
z;,  be  the  final  mean  velocity  of  the  water  in  f.  s. ; 
A  be  the  cross-sectional  area  of  the  stream  affected ; 
u  be  the  velocity  of  the  ship  in  f.  s. ; 
D  be   the   diameter   of   the   circle   described   by   the 
center  of  pressure  of  the   paddles  (this  point  may 

*  For  the  more  important  details  of  the  various  theories  which  have 
been  developed  along  the  lines  here  referred  to,  reference  may  be  made  to 
the  following: 

"  The  Mechanical  Principles  of  the  Action  of  Propellers."  By  Prof. 
Rankine.  Transactions  Institute  of  Naval  Architects,  vol.  vi.  p.  13. 

"  The  Minimum  Area  of  Blade  in  a  Screw  Propeller  necessary  to  Form 
a  Complete  Column."  By  Prof.  J.  H.  Cotterill.  Ibid.,  vol.  xx.  p.  152. 

"The  Part  played  in  the  Operation  of  Propulsion  by  Differences  in 
Fluid  Pressure."  By  R.  E.  Froude.  Ibid.,  vol.  xxx.  p.  390. 

"  The  Theoretical  Effect  of  the  Race  Rotation  on  Screw-Propeller  Effi- 
ciency." By  R.  E.  Froude.  Ibid.,  vol.  XXXTII.  p.  265. 

"  Marine  Propellers."     By  S.  W.  Barnaby.     London  and  New  York. 


PROPULSION. 


199 


be  taken  as  sensibly  at  the  center  of  the  paddle 
radially); 

N  be  the  number  of  revolutions  per  minute. 
Then  nDNA  is  the  volume  acted  on   per  minute  and  for 
sea- water:  6^.4nDNA  -f-  60  is  the  mass  in  pounds  acted  on 
per  second. 

Vi  —  ?'o  is  the  change  in  velocity  per  second.      Hence 


64.4*  DNA(v,  -  O 
thrust  =  T=          60x32.2          = 


nDNA 


The  thrust  equals  the   resistance,  and  hence,  solving  for  A, 
we  have 

A  = 


nDN(t\  —  v,)' 

In   this   expression    the   value    of  (i\  —  z/0)    is   uncertain, 
Most  writers,  following  Rankine,  have  considered  that 

vl  —  v0  —  slip  of  propeller  or  paddle-wheel  =  s— -? — . 

This  is  known  to  be  quite  inexact,  but  we  may  use  it  for  the 
present  illustrative  purpose.      We  have,  therefore, 

60  X 
7 

Or  we  may  put  more  generally  and  more  safely 

R 


(0 


And   likewise,  area   of   one   paddle   ~  A.     Hence  if  we 
denote  the  area  of  one  paddle  by  a,  we  may  put 

R 


a  ~ 


(2) 


200  RESISTANCE  AND    PROPULSION    OF  SHIPS. 

For  purposes  of  design  we  may  preferably  put  the  equa- 
tion in  another  form,  as  follows: 

We  have  R-^a(DNJs\    ......     (3) 

nDN(i-s) 

u  =  ~  TET  ~  ;     .....    <4) 

>.     Useful  work  or  Ru  ~  a(DN)ss(i  —  s).    .      .     (5) 

Also  Ru  ~  total  work  or  I.H.P. 
Denoting  this  by  /,  we  have 

f  ~  a(DNfs(i  -  s),      .....     (6) 


°r 


By  comparison  with  paddle-wheels  which  have  performed 
well,  constants  may  be  found  thus  relating  the  area  of  blade 
to  known  and  to  previously  assumed  quantities. 

For  radial  paddles  the  slip  s  may  be  taken  from  20  to  30 
per  cent,  and  for  feathering  paddles  from  15  to  20  per  cent. 

We  have  therefore 


The  value  of  K  will  depend  on  the  units  chosen  for  Z>', 
N'9  and  /.  For  numerical  convenience  we  may  take  these  as 
follows: 

/=  I.H.P.  absorbed  by  one  wheel; 
D'  —  (diam.  of  wheel  in  feet)  -f-  10  =  D  -+-  IO; 
N'  =  (revolutions  per  minute)  -f-  10  =  N  ~  10. 

We  may  then  take  K  from  2.0  to  2.5. 

This  fixes  the  area  of  one  float.     The  number  of  floats 


PROPULSION. 


201 


may  be  made  about  .8/7  for  radial  wheels,  and  from  .$D  to 
.jD  for  feathering  wheels.  The  proportions  of  floats  are 
usually 

Length  =  3  to  4  times  breadth. 

The  greatest  immersion  of  the  upper  edge  at  mean  draft 
should  be  from  .3  to  .8  the  breadth,  according  as  the  boat  is 
to  navigate  smooth  or  rough  water  and  to  vary  little  or  much 
in  draft. 

If  u  is  in  knots  per  hour,  instead  of  (4)  we  shall  have 

_  nDN(\  -s)  _  7tDN(\  -  s) 
Z  6080-- 60    :  101.3 


Hence 


DN  = 


(10) 


To  illustrate  these  formulae  take  the  following  data: 

/ '=  1000  on  both  wheels  and  500  on  one  wheel; 

u  =  1 8  knots, 

s  =  .20  per  cent,  assumed. 


Then 

and  D'N'  =  ; 

Hence,  taking  K  =  2.4, 

2.4  X  5QQ 
(7.255)'  X  .16 


a  •=. 


=  19.6  sq.  ft.,  or  say  20  sq.  ft. 


The  floats  might  therefore  be  8'  X  2'. 5. 

We  may  now  divide  DN  in  any  proportion  suitable  to  the 
circumstances  of  the  case.  Thus  if  we  take  N  '  —  40,  we  have 
D  —  1 8'.  i ;  or  if  we  put  D  =  15',  we  have  N  =  48.4. 


202  RESISTANCE  AND    PROPULSION   OF  SHIPS. 


Kinematic  Arrangement  of  Feathering  Paddles — In   Figs. 
61   and  62  are  shown  the  usual  kinematic  arrangements  for 


FIG.  62. 


feathering  paddles.     In  Fig.  61   the  links  similar  to  AB  are 
all  pivoted  at  A.     This  link  actuates  the  lever  BC  and  blade 


PROPULSION. 


203 


DE  as  shown,  and  similarly  for  the  others.  In  Fig.  62  HB 
is  a  drive-link  actuated  by  an  eccentric  with  center  at  A. 
The  remainder  of  the  links  are  pivoted  to  the  eccentric  strap 
as  shown,  and  are  connected  to  blades  similar  to  DE,  as  in 
Fig.  61.  In  both  diagrams  the  plane  of  the  blade  may  be 
beyond  the  pivot  C  as  shown,  or  it  may  contain  Cy  or  it  may 
lie  between  B  and  C.  The  arrangement  of  Fig.  62  is  more 
commonly  used,  that  of  Fig.  61  being  occasionally  employed 
for  small  wheels. 


39.  HYDRAULIC  PROPULSION. 

In  this  mode  of  propulsion  the  water  is  drawn  into  pumps 
or  other  similarly  acting  propelling  agents  situated  within  the 
boat.  It  is  then  acted  on  by  the  pump-vanes  and  delivered 
with  an  increased  velocity  toward  the  stern.  The  necessary 
reaction  is  thus  obtained  and  the  boat  is  moved.  This  mode 
of  propulsion  from  its  slight  use  does  not  merit  any  extended 
consideration.  We  may,  however,  instructively  glance  briefly 
at  certain  features. 

Let  M  be  the  water  acted  on  per  second  in  pounds.  For 
simplicity  we  may  assume  the  water  when  first  brought  under 
the  influence  of  the  propelling  agent  to  be  at  rest  relative  ta 
the  surrounding  still  water.  Let  ^,  be  its  velocity  sternward 
relative  to  the  same  datum  when  delivered  by  the  propelling 
apparatus.  The  change  of  velocity  per  second  is  then  #,. 
The  change  of  momentum  which  will  equal  the  thrust  will  be 
therefore 


T- 


Mv, 


204  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

The  useful  work  is  therefore 

w  __  TU  =  MVU 

-  gvj*» 

The  total  work  will  be  Tu  plus  that  involved  in  the  accel- 
eration and  disturbance  of  the  water,  and  possibly  in  raising 
it  against  gravity.  If  for  simplicity  we  neglect  the  loss  due 
to  eddies  and  disturbance  and  suppose  the  water  discharged 
under  the  same  average  head  as  that  under  which  it  enters, 
we  shall  have  for  the  total  work 


Hence  e  = 


This  is  sometimes  called  Rankine's  ideal  efficiency,  since 
it  is  the  limiting  value  of  the  efficiency  under  the  suppositions 
made.  In  the  actual  case  the  additional  losses  due  to  eddies, 


etc.,  will  raise  the  waste  work  to  -     -  or  more,  and  decrease 

o 

the  efficiency  to 

u 

e  =  -  -  -  or  less.  . 
u  +  v, 

Now  for  anything  approaching  moderate  or  high  speeds 
the  necessary  thrust  will  require  correspondingly  large  values 
of  M  and  vlt  or  of  both.  The  first  means  heavy  machinery 
and  the  second  low  efficiency.  In  any  practical  case  the 
value  of  vl  is  so  large  as  to  reduce  the  efficiency  below  that 
of  a  properly  installed  screw  propeller  or  paddle-wheel. 
Ideal  conditions  for  e  would  require  small  v^  and  hence  large 


PROPULSION. 


20$ 


M\  but  these  are  impossible  to  realize  in  practice,  and  the 
efficiency  is  therefore  necessarily  small. 

For  special  purposes  where  moderate  speed  is  sufficient, 
where  simplicity  is  an  important  element,  or  where  an  ordi- 
nary propeller  might  be  liable  to  race  violently  or  become 
fouled,  or  where  great  manoeuvring  power  is  required  as  in  a 
steam  life-boat,  this  mode  of  propulsion  may  have  advantages 
which  will  justify  its  use.  The  manoeuvring  power  is  usually 
obtained  by  multiple-discharge  nozzles  so  situated  and  so 
adjustable  that  the  reaction  may  be  directed  in  any  line,  and 
the  boat  propelled  in  either  direction  or  transversely,  or 
turned  in  either  direction  as  on  a  pivot. 


40.  SCREW  TURBINES  OR  SCREW  PROPELLERS  WITH 
GUIDE-BLADES. 

Thorny  croft's  screw  turbine  may  be  taken  as  an  illustra- 
tive example  of  this  type  of  propelling  agent.  The  propeller 
proper  consists  of  a  hub  ABC,  Fig.  63,  with  blades  ADEB 


very  much  longer  axially  and  shorter  radially  than  the 
common  propeller-blade.  In  consequence  the  inclination 
of  DE  to  the  axis  is  slight.  Aft  of  this  is  a  projection 
FGHKLMN  fixed  to  the  ship.  This  consists  of  a  prolonga- 
tion of  the  boss  by  the  spindle-formed  body  FGH,  a  cylin- 
drical shell  or  casing  KLMN,  and  a  series  of  blades  connecting 


2O6  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

this  casing  to  the  spindle-formed  body  as  indicated  at  KG 
and  HN.  These  blades  have  a  slight  inclination  in  the 
opposite  direction  to  that  of  the  propeller-blades  ADEB. 
The  pitch  of  the  latter  blades  is  variable.  The  product  of 
the  revolutions  by  the  pitch  at  E  is  intended  to  be  equal  to 
the  supposed  velocity  of  feed,  or  the  velocity  of  the  stern  of 
the  boat  through  the  wake  unaffected  by  the  action  of  the 
propeller.  The  product  of  the  revolutions  by  the  pitch  at  D 
is  likewise  intended  to  be  equal  to  the  speed  of  delivery,  and 
the  increase  of  size  in  ABC  and  consequent  reduction  in  area 
of  stream  are  intended  to  be  such  as  to  give  for  this  variable 
velocity  a  steady  flow  from  BE  to  DA.  The  water  on  leav- 
ing the  blades  of  the  propeller  has  a  considerable  rotary 
component,  due  to  which  it  impinges  on  the  fixed  blades 
KG.  By  their  guidance  it  is  brought  gradually  to  an  axial 
direction,  and  thus  finally  delivered.  The  long  projection 
FGH  is  for  the  purpose  of  providing  for  something  approach- 
ing steady  flow  after  leaving  the  guide-blades,  and  to  insure 
the  gradual  union  of  the  various  streams  with  each  other  and 
with  the  outside  water. 

From  the  character  of  the  actual  wake  as  explained  in 
§41,  it  is  quite  certain  that  while  in  a  general  way  the  inten- 
tion of  the  design  as  to  the  distribution  of  pitch  and  sectional 
area  of  stream  to  suit  the  velocities  of  feed  and  discharge  may 
be  partly  realized,  yet  such  result  must  be  far  from  exact,  and 
it  is  doubtful  if  such  variation  of  pitch  can  have  much  influ- 
ence on  the  actual  result. 

Remembering  the  relation  of  efficiency  to  the  character- 
istics of  a  propeller,  it  is  evident  that  there  is  no  reason  for 
expecting  any  marked  difference  in  this  feature  between  this 
form  and  the  common  propeller.  If,  then,  there  is  any  in- 


PROPULSION. 


2O7 


rease  in  ultimate  efficiency  it  will  probably  be  found  in  the 
relation  of  the  propelling  agent  to  the  ship.  Either  the 
augment  of  resistance  due  to  the  propeller  must  be  less  or  the 
wake  factor  greater  (§  44),  or  both.  There  seems  to  be  no 
reason  for  assuming  any  essential  difference  in  the  wake 
factor,  but  it  may  be  fairly  expected  that  due  to  the  pressure 
on  the  blades  KG,  the  normal  direction  of  which  will  have 
a  component  forward,  there  may  be  some  return  for  the 
increase  in  hull-resistance  due  to  the  action  of  the  propeller. 
On  the  other  hand  the  additional  attachment  furnishes  a  con- 
siderable increase  of  resistance,  and  hence  it  is  doubtful  if  the 
net  resistance  is  much  less  than  with  a  propeller  of  the  usual 
form.  We  should  therefore  expect  about  the  same  ultimate 
efficiency.  Such  conclusions  are  borne  out  by  experience. 

The  presence  of  the  surrounding  casing  does,  however, 
prevent  the  indraught  of  air  and  the  radial  escape  of  the 
water,  and  thus  increases  the  resistance  of  the  blades  to 
revolution,  and  hence  the  thrust.  It  results  that  the  requisite 
thrust  may  be  obtained  from  a  propeller  of  smaller  diameter 
than  when  of  the  usual  form,  especially  if  the  draft  is  small 
and  the  tips  of  the  blades  of  a  common  propeller  would  come 
near  the  surface  of  the  water.  In  such  cases  there  seems  to 
be  a  field  of  usefulness  for  propellers  of  this  character,  and 
many  of  the  Thornycroft  type  have  been  installed  with  very 
satisfactory  results. 


CHAPTER    III. 
REACTION    BETWEEN    SHIP   AND   PROPELLER. 

41.  THE  CONSTITUTION  OF  THE  WAKE. 

IN  our  treatment  of  propulsion  we  have  to  this  point 
assumed  the  propelling  agent  to  act  in  undisturbed  water. 
In  the  actual  case  this  is  far  from  correct,  and  we  have  now 
to  examine  the  effect  of  irregularly  disturbed  water  on  the 
preceding  results. 

We  will  first  note  the  cause  and  nature  of  the  disturbance. 
Taking  the  screw  propeller  as  the  typical  propelling  agent, 
its  place  of  action  is  at  the  stern  in  what  is  termed  the  wake, 
The  water  in  this  immediate  neighborhood  is  subject  to  the 
following  disturbing  causes: 

(i)  Stream-line  Motion. — In  §  2  we  have  seen  that  the 
relative  motion  between  the  water  and  the  ship  is  less  at  the 
stern  than  between  the  ship  and  the  outlying  undisturbed 
water,  so  that  relative  to  the  latter  the  stream-line  action 
will  give  a  motion  forward.  Due  to  the  same  general  cause, 
the  direction  of  the  relative  motion  of  the  water  and  ship  is 
aft,  with  an  upward  and  inward  component  as  it  follows  the 
contour  of  the  vessel.  This  brings  its  direction  of  flow 
oblique  to  the  plane  of  the  propeller  Due  to  the  difference 
in  their  location,  this  is  more  strongly  marked  with  twin- 
screws  than  with  a  single  screw.  The  influence  of  this 

obliquity  of  flow  will  be  discussed  at  a  later  point.     This  part 

208 


REACTION  BETWEEN  SHIP   AND    PROPELLER.        2OO, 

of  the  wake  is  frequently  referred  to  the  wave-motion  due  to 
the  system  of  waves  generated  by  the  motion  of  the  ship. 
We  prefer,  however,  to  refer  it  to  stream-line  motion,  as  a 
more  fundamental  aspect  of  the  cause.  This  naturally  in- 
cludes all  features  of  the  wake  contained  in  the  freely-flowing 
water,  or  in  that  not  involved  in  frictional  and  dead-water 
eddies. 

(2)  Skin  resistance  or  Frictional  Wake. — The  nature  of 
the  action  between  the  skin  of  the  ship  and  the  water  has 
been  discussed  in  §  7,  from  which  it  appears  that  the  whole 
surface  is  accompanied  by  a  layer  of  turbulent  water  partaking 
more  or  less  of  the  forward  movement.  This  motion  is 
greater  as  we  approach  the  after  end,  and  at  the  stern  we 
shall  have  a  considerable  mass  of  water  moving  forward  rela- 
tive to  the  surrounding  body  of  still  water. 

As  to  the  distribution  of  the  resulting  complex  wake,  we 
have  the  following  considerations: 

Near  the  surface  the  form  of  the  ship  is  relatively  full  and 
the  convergence  of  the  stream-lines  is  correspondingly  more 
marked.  The  disturbance  of  the  normal  horizontal  distribu- 
tion of  the  water  by  the  resulting  wave-motion  is  also  rela- 
tively more  marked  near  the  surface.  As  to  the  velocity  of 
the  frictional  wake,  there  seems  little  reason  to  expect  any 
marked  variation  with  depth  for  points  near  the  ship's  sur- 
face. The  velocity  will,  however,  rapidly  decrease  at  points 
successively  farther  and  farther  from  the  surface. 

As  to  the  actual  distribution  of  wake-velocity,  an  interest- 
ing series  of  experiments  has  been  carried  out  by  G.  A. 
Calvert.*  Across  the  stern  of  a  model  28  feet  long  and 
representing  a  full-bodied  cargo  steamer  was  fitted  a  frame 


*  Institution  of  Naval  Architects,  vol.  xxxiv.  p.  61. 


2IO  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

upon  which  were  stretched  several  vertical  wires  extending 
from  the  deck  to  some  distance  below  the  keel.  These  wires 
carried  Pitot  tubes  fitted  with  vanes  and  free  to  swing  to  the 
direction  of  flow.  An  outrigger  extending  into  undisturbed 
water  carried  a  similar  tube.  The  heights  recorded  in  these 
tubes  were  then  translated  into  velocities  by  the  usual 
formula,  v  =  V2gh. 

This  furnished  the  means  of  determining  the  velocity  of 
the  ship  relative  to  any  point  in  the  wake  and  relative  to  still 
water,  and  hence  the  velocity  of  the  water  at  this  point  of 
the  wake  relative  to  still  water.  The  wake-velocities  thus 
found  were  expressed  as  percentages  of  the  velocity  relative 
to  still  water.  In  the  section  of  the  wake  including  the 
screw  aperture,  the  boss  alone  of  the  screw  being  fitted,  the 
percentages  in  a  vertical  line  near  the  stern-post  varied  from 
64  to  17  per  cent  from  the  surface  downward;  in  a  horizontal 
line  near  the  surface,  from  64  to  7  per  cent  from  the  stern- 
post  outward;  and  in  a  horizontal  line  near  the  keel,  from  17 
to  10  per  cent  from  the  stern-post  outward.  The  average 
amount  at  this  section  was  about  19  per  cent. 

An  attempt  was  next  made  to  determine  the  portion  of 
this  due  to  the  frictional  wake.  To  this  end  a  thin  plank  of 
the  same  length  as  the  model  was  towed  at  the  same  speeds. 
Similar  measuring  appliances  being  fitted,  the  wake  speeds  at 
points  near  the  surface  of  the  plank  at  distances  of  I,  7,  14, 
21,  and  28  feet  from  the  forward  end  were  found  respectively 
16,  37,  45,  48,  and  50  per  cent  of  the  speed  of  the  plank. 
It  was  also  shown  that  approximately  the  wake-velocities 
decreased  in  a  geometrical  progression  as  the  distances  from 
the  surface  increased  in  arithmetical  progression. 

From  various  considerations,  for  the  details  of  which  the 


REACTION  BETWEEN  SHIP   AND    PROPELLER.        211 


original  paper  may  be  consulted,  the  author  concludes  that  of 
the  19  per  cent  average  wake  about  5  per  cent  was  due  to 
frictional  wake,  9  per  cent  to  wave-motion,  or  stream-line 
motion  as  we  prefer  to  term  it,  and  the  remaining  5  per  cent 
unaccounted  for  otherwise  is  charged  to  the  influence  of 
"  dead  water"  or  eddying  water  about  the  stern  due  to  the 
comparative  fullness  of  form. 

It  thus  appears  that  the  wake-velocity  in  general  is  ex- 
ceedingly variable  throughout  the  cross-section  of  the  wake 
stream.  Also  from  this  and  other  experimental  determina- 
tions to  be  referred  to  at  a  later  point,  it  appears  that  includ- 
ing simply  the  water  immediately  astern  of  the  ship,  and 
hence  that  likely  to  be  influenced  by  propellers,  its  average 
value  may  vary  from  6  or  8  per  cent  to  20  or  25  per  cent  of 
the  speed  of  the  vessel.  In  general  it  is  found  that  the  wake 
value  is  greater  as  the  ship  is  longer  and  fuller,  and  less  as  it 
is  shorter  and  finer.  For  empirical  equations  relating  its 
value  to  the  characteristics  of  the  ship  see  §  50,  (9)  and  (10). 

42.  DEFINITIONS  OF  DIFFERENT  KINDS  OF  SLIP,  OF  MEAN 
SLIP,  AND  OF  MEAN  PITCH. 

It  will  be  remembered  that,  'unless  otherwise  stated,   all 
velocities  are  referred  to  the  surrounding  body  of  still  water 
as  datum.     We  will  first  suppose  the  wake  to  be  uniform. 
Let  u  =  speed  of  ship  ; 

z>0  =  speed  of  wake  if  propeller  were  not  acting  and 

the  ship  were  towed  at  the  speed  u ; 
z/,  =  actual  speed  of  water  at  or  just  forward  of  the 

propeller; 

vt  =  final  speed  of  water  due  to  influence  of  the  pro- 
peller; 


212  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

v  =  pitch  X  revolutions  =  pN  =  speed  of   advance 
of  propeller  through  wake   if  there  were   no 
slip. 
Then  u  —  v0  =  speed  of  advance  of  propeller  relative  to  the 

wake; 

u  —  vl  =  speed  of  advance  of  propeller  relative  to  the 
water  immediately  about  it. 

v  —  u  =  apparent  slip  of  propeller  =  S^ ;       .      .      .      (i) 
v  —  (u  —  v0)  =  (v—  u)-\-v^=Sl-}-v0  =  tr2ie  slip  of  propeller—  S2 ; 

or      5,  -f-  VQ  =  true  slip  of  propeller  —  52; (2) 

v0  —  vt  =  total  slip  of  water  =  53 (3) 

Usually  v0  is  directed  forward  and  v^  aft,  and  hence  53  will 
be  numerically  the  sum  of  v0  and  v9. 

It  should  be  especially  noted  that  these  three  kinds  of  slip 
are  separate  and  distinct,  though  of  course  not  independent. 
In  particular  it  should  be  noted  that  5a  and  53  are  not  the 
same. 

In  the  simple  ideal  case  of  Fig.  55  we  may  readily  relate 
the  value  of  58  to  52  or  ja  as  follows: 

It  is  shown  in  mechanics  that  the  action  of  a  surface  in 
deflecting  a  stream — omitting  skin-resistance  and  the  forma- 
tion of  eddies — involves  simply  a  change  in  direction  without 
change  in  velocity,  and  the  total  force  interaction  between  the 
stream  and  the  surface  is  measured  in  direction  and  amount 
by  the  resultant  change  in  momentum.  In  that  diagram  the 
water  is  considered  as  approaching  the  blade  with  velocity 
represented  in  amount  and  direction  by  EA,  and  as  leaving 
with  the  same  velocity  parallel  to  the  blade.  This  is  repre- 
sented in  amount  and  direction  by  making  KA  =  EA.  Then 
by  the  usual  composition  of  motions  the  resultant  change  is 


REACTION  BETWEEN  SHIP   AND    PROPELLER.        213 

represented   in   direction  and   amount   by  EK.     The   longi- 
tudinal component  of   this  is  EG,  and  this   is  therefore  the 
acceleration  which  is  directly  useful  in  providing  thrust. 
Now  if  the  slip  angle  CAR  is  small  we  may  put 


EG  —  EC  cos3  a  —  s^pN  cos3  a  =  5a  cos3  a. 


(4) 


It  thus  appears  that  the  velocity  S8  is  variable  for  a  given 
value  of  Sa,  increasing  as  a  decreases,  and  therefore  increas- 
ing from  the  hub  outward,  at  all  points,  however,  being  less 
than  Sv 

Effect  on  the  Preceding  due  to  the  Irregularity  of  the  Wake. 
— From  (2)  we  have 


whence 


(5) 


Now  remembering  the  actual  wake  as  described  in  §  41  it 
appears  that  v0  -r-  v  may  vary  from  perhaps  o  to  50  per  cent 
or  more.  The  value  of  sl  is  usually  found  between  10  and 
20  per  cent.  Hence  s^  may  vary  from  say  5  or  10  per  cent 
to  70  per  cent  or  perhaps  even  more.  It  thus  appears  that 
the  value  of  the  true  slip  is  variable  over  the  surface  of  the 
blade  and  from  point  to  point  in  the  revolution  between  the 
very  wide  limits  noted  above,  and  these  may  not  perhaps  be 
the  widest  extremes.  It  is  indeed  quite  possible  that  for 
certain  elements  the  slip  might  be  o  or  even  negative,  while 
for  others  its  value  might  rise  possibly  to  nearly  100  per  cent. 
It  is  thus  seen  that  the  idea  of  slip  for  the  propeller  as  a 
whole,  acting  in  the  wake,  has  lost  entirely  the  simplicity  of 
meaning  which  we  were  able  to  give  to  it  when  dealing  with 
a  single  element  in  undisturbed  water. 


214  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

In  spite,  however,  of  the  wide  range  of  extreme  values 
which  it  seems  possible  that  the  slip  may  have,  the  conditions 
for  most  of  the  blade  and  for  most  of  the  revolution  are  such 
as  to  give  rise  to  a  much  narrower  range  of  fluctuation,  and 
most  of  the  work  is  undoubtedly  done  between  a  compara- 
tively narrow  range  of  variation.  We  are  thus  led  to  the 
definition  of  mean  slip.  This  admits  of  definition  in  various 
ways. 

(a)  It  might  be   taken  as  the   arithmetical  mean  for  an 
entire  revolution,  of  the  various  slips  of  the  elements  compos- 
ing the  driving-face. 

(b)  Since  these  elementary  values  are    quite    unequal  in 
relative  importance  according  to  their  location  and  the  value 
of  the  slip,  it  might  seem  more  fair  to  give  to  each  element  a 
weight  corresponding  to  the  gross  work  absorbed  by  it.     Thus 
if  w^  w^  w^  etc.,  denote  the  amount  of  work  absorbed  by 
each  element,  and  slt  s^  s3,  etc.,  the  corresponding  values  of 
the  slip,  then 

mean  slip  = 


i .{_  Wii  _|_  .  w  ' 

(c)  Instead  of  giving  to  each  elementary  slip  a  weight 
proportional  to  its  gross  work,  we  may  take  its  thrust  or  use- 
ful work.  Denoting  the  elementary  thrusts  by  /,,  /„  etc., 
we  should  then  have 


mean  shp  = 


An  approximate  method  of  applying  the  weights  thus  indi- 
cated in  (b)  and  (c)  will  be  given  below  under  the  discussion 
of  mean  pitch. 

(</)  In  §  35,  (12),  the  total  work  of  an  element  is  expressed 


REACTION  BETWEEN  SHIP  AND    PROPELLER.        21$ 

as  a  function  of  the  geometrical  form  of  the  propeller  and  the 
slip.  For  the  entire  propeller  with  variable  slip  the  total 
work  will  be  represented  by  the  summation  of  such  elements. 
Suppose  now  that  the  slip  instead  of  variable  is  constant  at 
a  value  s,  all  other  conditions  remaining  as  before,  and  that 
the  corresponding  summation  for  total  work  W  is  the  same 
as  before.  Then  evidently  s  may  be  considered  relative  to 
the  variable  distribution  of  slip  as  a  mean  or  equivalent 
value.  This  is  the  same  as  defining  mean  slip  as  the  slip  at 
which  the  same  propeller  in  a  uniform  stream  at  the  same 
number  of  revolutions  would  absorb  the  same  total  work  as 
in  the  actual  case. 

(e)  Instead  of  total  work  as  in  (</),  the  definition  may  be 
similarly  founded  on  useful  work  or  thrust.  This  is  the  same 
as  defining  mean  slip  as  the  slip  at  which  the  same  propeller 
in  a  uniform  stream  at  the  same  number  of  revolutions  would 
give  the  same  useful  work  or  the  same  thrust  as  in  the  actual 
case. 

In  all  of  our  references  thus  far  to  an  entire  propeller  we 
have  assumed  it  to  be  of  uniform  pitch  on  the  driving-face. 
On  this  assumption  we  have  discussed  the  variability  of  slip 
over  the  surface  and  throughout  the  revolution,  and  have 
given  various  definitions  of  mean  slip  as  above.  Propellers 
are  frequently  made,  however,  with  pitch  variable  over  the 
driving-face,  and  thus  is  introduced  another  element  of  varia- 
tion into  the  distribution  of  slip,  and  also  the  need  for  some 
definition  of  the  terms  mean  pitch  and  mean  slip  as  applied  to 
such  propellers. 

For  the  former  we  may  take  a  geometrical  basis  and  define 
mean  pitch  as  the  mean  of  the  distributed  values  taken  over 
the  driving-face.  Where  the  pitch  varies  simply  from  the 


2l6 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


leading  to  the  following  edge,  the  mean  of  the  two  values  at 
these  points  is  frequently  taken  as  the  mean  pitch  instead  of 
the  more  distributed  mean  as  above. 

Instead  of  taking  the  simple  mean  of  the  distributed 
values,  we  might  perhaps  more  properly  give  to  the  pitch  of 
each  element  a  weight  proportional  to  the  work  which  it 
absorbs  or  to  the  thrust  which  it  develops.  By  reference 
to  §  35,  (8)  and  (6),  it  appears  that  this  would  be  very  closely 
realized  by  giving  to  the  pitch  of  each  element  a  weight 
represented  in  the  first  case  by  the  product  of  its  area  by  the 
cube  of  its  velocity  or  by  the  cube  of  its  radius,  and  in  the 
second  case  by  the  product  of  the  area  by  the  square  of 
the  velocity  or  radius.  The  following  actual  case  of  a  four- 
bladed  model  propeller  may  be  given  as  an  illustration  of  the 
determination  of  the  reduced  mean  pitch  in  the  second 
manner.  The  mean  pitch  across  the  blades  was  taken  at  four 
radial  distances  nearly  equal  to  .3,  .5,  .7,  and  .9  the  radius, 
and  the  area  was  taken  as  proportional  to  the  breadth.  The 
work  for  one  blade  was  then  arranged  as  follows: 


Radius. 

r* 

Breadth. 

Pitch. 

r*l> 

rHp 

i-77" 
2.97 
4.07 
5.26 

3-13 
8.82 
16.56 
27.67 

3-30" 
3-60 
3-30 
2.16 

16.13" 
15.90 
15.96 
15-78 

10.33 
31-75 
54-65 
59-76 

166.6 
504.8 
872.2 
943-2 

156.49 

2486.8 

15.89 

The  quotient  of  the  sum  of  the  column  r*b  into  the  sum 
r*bp  gives  for  this  blade  the  reduced  mean  pitch  equal  to 
15.89".  For  the  other  blades  the  values  similarly  found 


REACTION  BETWEEN  SHIP   AND   PROPELLER.        21 J 


were  15.45",  15.80",  and  15.54".  This  gives  as  the  final 
mean  for  the  entire  propeller  the  value  15.67". 

A  value  of  the  mean  pitch  being  thus  found  from  either  a 
simple  or  weighted  mean,  a  definition  of  mean  slip  follows 
thus: 

Assume  a  propeller  similar  to  the  one  given  in  all  respects 
except  as  to  pitch,  which  shall  be  uniform  and  of  the  mean 
value  as  above  defined,  and  let  this  propeller  working  in  a 
uniform  stream  at  the  same  number  of  revolutions  absorb  the 
same  amount  of  total  work  as  the  given  propeller.  Then  the 
slip  at  which  such  conditions  would  be  fulfilled  would  be  a 
mean  or  equivalent  slip  for  the  given  propeller.  This  is  evi- 
dently similar  to  (d)  above  for  the  case  of  uniform  pitch. 
We  may  also  as  in  (e)  take  the  useful  work  or  thrust  as  the 
basis  for  a  similar  definition  of  mean  slip. 

Instead  of  defining  mean  pitch  on  a  purely  geometrical 
basis,  we  may  give  it  a  dynamical  definition  as  follows: 

Let  the  given  propeller  work  in  undisturbed  water  with 
given  revolutions  and  speed.  Let  there  be  a  propeller  of 
uniform  pitch  with  the  same  diameter,  area,  and  shape  of 
blades,  and  let  it  work  in  undisturbed  water  at  the  same 
revolutions  and  speed.  Then  the  pitch  at  which  the  latter 
propeller  would  have  the  same  turning  moment,  or  at  which 
it  would  absorb  the  same  work,  as  the  first,  may  be  consid- 
ered as  the  equivalent  mean  pitch  of  the  former.  Similarly 
the  definition  may  be  based  on  equivalent  thrusts  instead  of 
equivalent  turning  moments. 

It  may  be  remarked  that  the  pitch  thus  determined  would 
doubtless  vary  with  the  revolutions,  and  with  the  relation 
between  revolutions  and  speed,  so  that  it  could  not  be  con- 
sidered as  a  fixed  dimension  of  the  propeller. 


218 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


A  dynamical  definition  of  mean  pitch  having  been  thus 
taken,  the  definition  of  mean  slip  would  most  naturally  follow- 
according  to  (d)  or  (e)  above. 

In  all  such  cases  where  the  mean  pitch  or  slip  of  a  pro- 
peller is  based  on  its  performance  in  comparison  with  that  of 
a  propeller  of  uniform  pitch,  the  influence  of  the  thickness  or 
rounded  back  is  virtually  involved.  To  this  we  may  briefly 
allude. 

Given  a  body  of  the  cross-section  of  a  propeller  blade  as 
in  Fig.  64,  moved  relative  to  the  water  in  the  direction  of  the 


FIG.  64. 

face  BA.  Then  the  distribution  of  the  stream-lines  is  such 
that  relative  to  the  pressure  on  AB  there  is  an  excess  from 
A  to  C  and  a  more  or  less  pronounced  defect  between  B 
and  C.  The  result  is  in  general  represented  by  a  pressure  on 
the  face  AB  having  its  center  much  nearer  B  than  A,  and 
hence  tending  to  turn  the  blade  about  in  the  clockwise  direc- 
tion as  here  viewed.  If  the  direction  KL  is  axial,  there  will 
result  in  such  case  a  positive  or  forward  thrust.  That  is,  if 
the  propeller  be  run  in  undisturbed  water  at  such  revolutions 
relative  to  the  speed  that  the  slip  of  the  driving-face  is  o, 
there  will  in  general  result  a  slight  positive  or  forward  thrust 


REACTION  BETWEEN  SHIP   AND    PROPELLER. 

due  to  this  distribution  of  the  stream-line  motion,  and  it  is 
not  until  the  slip  of  the  driving-face  is  still  further  decreased 
and  made  slightly  negative  that  the  fore  and  aft  components 
of  the  total  surface  forces  exactly  balance  and  give  a  zero 
thrust.  It  may  also  be  noted  that  this  zero  thrust  is  actually 
the  resultant  of  a  component  aft  due  to  the  tangential  forces 
or  those  due  to  skin-friction  and  edge-resistance,  and  of  a 
component  forward  due  to  the  normal  forces  or  those  due  to 
the  stream-line  pressures.  If,  therefore,  the  former  could  be 
reduced  or  eliminated  the  propeller  would,  under  the  condi- 
tions just  assumed,  still  show  a  positive  thrust,  and  it  would 
require  a  still  further  increase  of  negative  slip  to  reduce  the 
longitudinal  component  of  the  normal  pressures  to  zero. 
These  results  have  been  frequently  noted  experimentally,  and 
have  been  made  the  subject  of  quantitative  measurement  in 
a  series  of  experiments  carried  on  by  the  author.* 

This  is  what  is  frequently  referred  to  by  the  statement 
that  the  addition  of  thickness  results  in  a  virtual  increase  of 
pitch,  because  if  the  equivalent  pitch  be  taken  as  (longi- 
tudinal speed  for  zero  thrust)  -f-  (revolutions),  the  result  will 
be  greater  than  that  derived  by  the  measurement  of  the  driv- 
ing-face, and  the  excess  would  be  still  greater  could  the 
tangential  forces  be  eliminated. 

We  have  thus  discussed  various  possible  definitions  of 
mean  pitch  and  mean  slip  in  order  to  show  the  variety  of 
meaning  which  may  be  given  to  these  terms,  and  the  conse- 
quent inexactness  of  significance  attending  their  use  without 
some  agreement  as  to  the  basis  of  definition.  Our  use  of  the 
equations  of  §  36  will  virtually  assume  the  definition  of  mean 
slip  as  based  on  (e).  That  is,  we  shall  assume,  as  will  be 

*  Transactions  Society  Naval  Architects  and  Marine  Engineers,  vol.  V. 


220  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

explained  later,  that  for  the  actual  variable  slip  may  be  sub- 
stituted some  constant  equivalent  value  for  use  in  the  equa- 
tions giving  the  value  of  the  total  useful  work. 

In  regard  to  propellers  of  variable  pitch  it  may  be  noted 
that  such  design  is  usually  intended  to  provide  for  some 
variable  distribution  of  slip  over  the  surface,  assuming  the 
propeller  to  work  in  a  uniform  stream.  When,  however,  we 
remember  the  great  variability  of  the  stream  or  wake,  it  is 
quite  evident  that  any  attempt  to  secure  any  specified  distri- 
bution would  be  entirely  futile,  and  that  in  any  given  case  the 
actual  distribution  will  be  quite  different  from  that  intended. 
Hence  any  effects  resulting  from  a  variable  pitch  will  be  quite 
accidental,  and  it  is  very  doubtful  if,  in  the  present  state  of 
our  knowledge,  there  is  anything  to  be  gained  by  introducing 
such  a  feature  into  our  designs.  These  conclusions  seem  to 
be  borne  out  by  experience,  for  propellers  of  uniform  pitch 
have  shown  themselves  in  practice  to  be  equal  in  efficiency  to 
those  in  which  the  pitch  is  variable  according  to  various  laws. 
We  shall  therefore  pay  no  further  attention  to  variable  pitch 
as  a  feature  of  screw  propellers,  but  shall  in  all  cases  assume 
them  to  be  of  uniform  pitch  over  the  entire  driving-face. 

43.  INFLUENCE  OF  OBLIQUITY  OF  STREAM  AND  OF  SHAFT 
ON  THE  ACTION  OF  A  SCREW  PROPELLER. 

In  general  the  line  of  the  shaft,  the  direction  of  the  stream, 
and  the  direction  of  advance  are  all  different.  This  is  illus- 
trated in  Fig.  65,  where  AC  represents  the  direction  and 
speed  of  advance  of  the  ship  relative  to  still  water,  BC  the 
direction  and  speed  of  the  stream  relative  to  the  same,  OF 
the  direction  of  the  shaft,  and  OA  the  speed  and  direction 
of  the  element  relative  to  the  ship.  Then  by  the  composi- 


REACTION  BETWEEN  SHIP  AND    PROPELLER.        221 

tion  of  relative  motions  it  follows  that  BA  represents  the  direc- 
tion and  speed  of  the  stream  relative  to  the  ship,  and  BO  that 
of  the  stream  relative  to  the  blade  or  element.  Let  z>0  denote 
the  velocity  of  the  stream  relative  to  still  water  as  represented 


FIG.  65. " 

by  BC,  and  denote  the  angle  BCA  by  rj.  Then,  as  in  §  33, 
we  denote  the  angle  between  OG  and  the  normal  ON  to  the 
element  by  a.  The  inclination  of  z/0  to  the  normal  is  then 
(a  —  ij),  and  v9  cos  (a  —  77)  is  the  component  of  t/0  in  the 
direction  of  the  normal,  and  hence  the  velocity  in  this  direc- 
tion which  would  be  impressed  on  the  element  by  the  stream 
of  velocity  vof  if  there  were  no  slip.  The  result  of  this  in  the 
direction  of  advance  would  be  a  velocity  v9  cos  (a  —  rj)  sec  a. 
From  §  33  the  longitudinal  velocity  without  slip  in  still 
water  would  be  v  cos  /3  sec  a.  Hence  denoting  the  total 
speed  of  advance  if  there  were  no  slip  by  «',  we  shall  have 

u'  =  v  cos  ft  sec  a  -|-  v9  cos  (a  —  rf)  sec  a 

=  (y  cos  fi  -f-  v9  cos  (a  —  77))  sec  a. 

In  the  present  case,  with  the  shaft  at  an  angle  e  with  the 
line  of  advance,  the  angle  fi  for  a  given  element  is  constant, 


222  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

while  a  varies.  The  exact  values  of  a  depend  on  the  solution 
of  a  spherical  triangle,  as  is  readily  seen,  and  these  values  will 
vary  through  a  range  2e.  In  consequence,  the  values  of  the 
speed  of  advance  without  slip,  u' ',  will  vary  through  a  corre- 
sponding range.  Denote  the  mean  value  of  «'  by  #/  and  the 
range  of  its  variation  by  2Au' .  Also,  let  6  denote  the  angular 
location  of  the  element  reckoned  from  the  position  for  which 
u'  is  maximum.  Then  it  is  found,  if  e  and  VQ  are  relatively 
small,  that  we  have  approximately 

u'  =  #/  +  Au'  cos  0. 

The  value  of  u'  thus  given  is  seen  to  vary  from  uj  +  Au' 
through  «/,  uQf  —  Au' ,  u0  to  #/  +  Au'  for  the  successive 
quadrants  of  an  entire  revolution.  The  actual  speed  of 
advance  being  &,  it  follows  that  the  difference  u'  —  u  or  the 
actual  slip  5,  and  the  percentage  slip  s  may  likewise  be 
expressed  in  the  same  approximate  form.  Hence  we  shall 
have 

£**   O       I        /f  C*  /3 


s  =  s0  +  As  cos  6. 

Denote  the  mean  value  of  a  by  aQ.  This  is  seen  to  be 
the  angle  between  the  normal  and  the  direction  of  the  shaft. 
Then 

u0'  =  (v  cos  /3  -f-  v0  cos  (a0  —  ij) )  sec  a0 ; 

and  S0  =  u0'  —  u. 

We  must  now  find  the  value  of  Au'  =  AS.  We  have  for 
the  maximum  value  of  u'  approximately 

^max.  =  (V  COS  ft  +  V0  COS  (flf0  +  €  -  ?)  )  SCC  (flf0  +  e). 

Then  z/max.  -  «/  =  Au'  =  AS. 


REACTION  BETWEEN  SHIP   AND    PROPELLER.        22$ 

Considering  e  small,  this  difference  is  readily  put  in  the 
following  approximate  form : 

Au'  =  AS  =  u0r 


cot  a0  —  e 

It  thus  appears  that  the  values  of  AS  and  hence  of  As 
vary  over  the  entire  surface  of  the  blade,  and  hence  the  entire 
distribution  of  5  and  s  will  vary  from  point  to  point  over  the 
surface  and  from  one  position  to  another  in  the  revolution. 
The  value  of  the  range  of  slip  2AS  is  seen  to  increase  with 
e  and  with  a0.  Hence  the  value  of  AS  will  continuously 
decrease  for  locations  of  the  element  from  the  hub  outward. 
The  position  from  which  6  must  be  counted  is  seen  from  the 
following  considerations.  The  origin  for  0  is  the  prositon  for 
maximum  value  of  #';  and  since  VQ  is  small,  u'  will  depend 
chiefly  on  v,  and  will  reach  its  maximum  very  near  the  posi- 
tion for  which  sec  a  is  maximum  or  a  maximum  or  when 
a  =  aQ  -f-  e.  Hence  the  o  position  of  0  is  readily  seen  to  be 
near  that  in  which  the  normal  is  parallel  to  the  plane  deter- 
mined by  the  shaft  and  direction  of  advance,  and  in  which 
the  normal  and  direction  of  advance  lie  on  opposite  sides  of 
the  shaft. 

Thus  with  twin  screws,  the  starboard  turning  to  the  right 
and  the  port  to  the  left,  let  the  shafts  incline  outward  from 
the  engine  aft,  as  usually  fitted.  Then  it  follows  from  the 
above  that  for  each  blade  of  each  propeller  the  slip  is  maxi- 
mum near  its  highest  position  and  minimum  near  its  lowest. 
Similarly,  with  a  right-hand  propeller  and  the  shaft  inclined 
downward  the  slip  of  any  blade  is  maximum  when  it  is  hori- 
zontal and  directed  to  the  right,  and  minimum  when  hori- 
zontal and  directed  to  the  left.  With  right-  and  left-hand 


274  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

twin  screws,  as  above  noted,  the  effect  of  the  obliquity  of  the 
shafts  is  still  further  increased  by  the  natural  obliquity  of  the 
inflowing  streams  as  they  follow  the  contour  of  the  ship,  thus 
giving  a  direction  of  stream  relative  to  still  water  somewhat 
like  BC  of  Fig.  65.  Inclination  of  the  shafts  in  the  opposite 
direction  would  tend  to  correct  the  variability  of  the  slip  due 
to  these  streams,  but  from  structural  reasons  this  is  rarely 
permissible. 

We  have  discussed  in  the  preceding  section  the  influence 
of  the  variability  of  the  wake  on  slip,  and  in  the  preseht  sec- 
tion the  influence  of  obliquity,  assuming  thus  far  a  uniform 
stream  or  wake.  Combining  these  influences  we  shall  evi- 
dently have  in  any  given  case  variations  of  slip  over  the 
surface  and  throughout  the  revolution  ranging  through  limits, 
variable  themselves,  but  probably  as  wide  as  from  o  or  a 
negative  value  to  50  or  75  per  cent  or  more.  We  shall  have 
therefore  the  variability  noted  in  §§  41,  42,  with  an  added 
amount  due  to  obliquity.  As  noted  in  §  41,  however,  most 
of  the  work  will  be  done  between  narrower  ranges,  but  due 
to  obliquity  even  these  may  be  considerable.  It  becomes 
therefore  a  matter  of  importance  to  inquire  to  what  extent 
the  formulae  of  §  36  may  be  invalidated  by  the  existence  of 
this  variability  of  slip. 

44.  EFFECT  OF  THE  WAKE  AND  ITS  VARIABILITY  ON  THE 
EQUATIONS  OF  §  36. 

We  will  first  take  the  influence  of  the  variability  of  the 
wake  on  the  efficiency. 

Let  eiy  €„  etc.,  denote  the  efficiencies  of  the  various 
elements.  Then  using  the  previous  nomenclature  we  shall 
have  for  the  entire  propeller 


REACTION  BETWEEN  SHIP   AND   PROPELLER. 
U 


225 

(0 


If  now  the  value  of  ^  varies  by  a  linear  law  with  slip  for 
the  range  of  variation,  we  shall  have 

e  =  e       as 


•  =  ^o  + 
etc. 


Hence  we  find 


,s,  4-  w,s, 


,  +  w,  + 


We  know  that  efficiency  does  not  actually  vary  by  a  linear 
law  with  slip,  but  if  most  of  the  work  is  done  between  a 
relatively  narrow  range  of  variation  of  slip,  the  variation  of 
efficiency  may  be  approximately  expressed  by  such  a  law. 
The  resulting  efficiency  in  (2)  is  then  seen  to  be  that  corre- 
sponding to  the  mean  slip  as  defined  in  §  42  (&).  The  actual 
efficiency  will  be  somewhat  less  than  that  given  by  (2),  and 
we  may  consider  that  this  equation  simply  indicates  the 
general  conditions  under  which  no  great  loss  of  efficiency 
shall  result  from  variable  slip,  viz.,  that  most  of  the  work 
must  be  done  within  a  comparatively  narrow  range  of  slip 
variation. 

This  conclusion  is  borne  out  by  the  experiments  of  R.  E. 
Froude,  and  we  may  consider  that  the  latter  constitute  really 
the  basis  for  our  assumptions  relative  to  the  wake.  These 
are: 

That  the  actual  turbulent  variable  wake  may  be  consid- 
ered as  sensibly  equivalent  to  a  single  uniform  wake,  and  that 


226  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

the  performance  of  the  propeller  may  be  considered  as  equiva- 
lent to  that  which  would  result  from  the  presence  of  such  a 
wake. 

The  true  mean  slip  5a  is  then  greater  than  the  apparent 
slip  S1  by  the  amount  of  this  uniform  substituted  wake,  which 
we  may  denote  by  VQ.  Vice  versa  it  results  that  the  amount 
of  the  uniform  substituted  wake  is  equal  to  the  difference 
between  the  true  mean  slip  based  on  the  definition  of  §  42, 
(e),  and  the  apparent  slip  5,. 

Stating  again  the  relation  somewhat  differently,  we  note 
that  the  true  slip  is  greater  by  variable  amounts  than  the 
apparent  slip,  and  the  actual  thrust  developed  is  greater  than 
that  which  would  result  in  undisturbed  water  by  the  screw 
working  at  the  apparent  slip.  At  some  greater  slip  in  undis- 
turbed water,  however,  the  thrust  developed  would  be  the 
same  as  that  actually  obtained.  This  greater  slip  may  then, 
according  to  §  42  (e)  be  considered  as  the  equivalent  or  mean 
true  slip  in  the  actual  case,  and  the  difference  between  the 
true  and  apparent  slips  thus  defined  will  represent  the  amount 
of  uniform  wake  considered  as  equivalent  to  the  actual  vari- 
able wake. 

This  definition  of  equivalent  wake  is  based  on  an  equiva- 
lence of  thrusts,  as  stated  in  §  42  (e).  We  might  also  base 
a  definition  on  an  equivalence  of  turning  moments  or  total 
works,  as  stated  in  definition  (d).  If  now  the  wake  itself 
were  uniform  these  two'  definitions  or  modes  of  determination 
would  evidently  lead  to  the  same  value  of  the  wake.  The 
difference  in  the  two  values  thus  determined  indicates  there- 
fore the  effect  due  to  the  turbulence  or  irregularity  of  the 
wake.  Experiments  on  this  point  are  not  sufficiently  ex- 
tended to  furnish  very  complete  evidence  as  to  the  exact 


REACTION  BETWEEN  SHIP  AND    PROPELLER.        227 

influence  which  turbulence  may  play,  but  its  value  is  believed 
to  be  small,  and  in  any  event  in  default  of  sufficiently 
extended  information  we  are  compelled  to  assume  its  influ- 
ence as  negligible.  The  actual  point  where  the  influence  of 
turbulence  or  irregularity  of  wake  touches  our  methods  of 
design  is  in  its  influence  on  efficiency,  to  which  reference  has 
been  made  in  the  early  part  of  the  present  section. 

We  therefore  virtually  assume  in  all  cases,  for  the  actual 
turbulent  wake,  the  substitution  without  change  in  efficiency, 
of  a  uniform  wake  of  velocity  z/0,  the  amount  of  this  velocity 
being  based  on  an  equivalence  of  thrusts  as  previously  ex- 
plained. Let 


u  —  vn 


=  w. 


This  relates  v^  to  (u  —  v0\  the  speed  of  advance  of  the  pro- 
peller through  the  water  about  the  stern.     Whence 

wu 

0  ~~  I  +  w  ' 

From  §  42,  (5),  we  have 

,     V,  W       U 


W 


whence 


w  = 


i  — 


(3) 


I   ~f-  W  = 


I  —  s 


(4) 


228  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

Let  us  next  consider  the  influence  of  the  uniform  wake 
on  the  expressions  for  useful  work  and  efficiency  in  §  36,  (2) 
and  (4). 

It  is  readily  seen  that  the  useful  work  must  be  obtained 
by  multiplying  the  thrust  by  the  speed  of  the  ship  relative  to 
still  water,  and  hence  by  (i  —  s^pN.  In  all  other  places, 
however,  where  slip  enters  into  these  expressions  it  will  be 
the  true  slip  st.  Hence  we  shall  have  in  such  case 


U  =  7T3/W3(i  -  s^y\asB  -  fC)dA  ; 
W  =  n*p*N*y\asB  +  fE}dA  . 


The  ratio  of  these  two  will  give  an  apparent  efficiency. 
The  true  efficiency  must,  of  course,  be  that  for  the  propeller 
working  at  a  slip  of  s^.  The  presence  of  the  wake  cannot 
change  the  efficiency  of  the  propeller  itself,  while  it  may 
increase  the  amount  of  useful  result  U.  The  increase  in  U 
must  be  credited  to  the  wake  rather  than  to  the  propeller. 
These  points  will  be  again  referred  to  in  §  46.  For  these 
reasons  we  call  the  ratio  of  U  to  W  an  apparent  rather  than 
a  real  propeller  efficiency.  This  we  may  denote  by  elt  while 
the  true  efficiency,  which  we  will  denote  by  e,t  will  have  the 
value  as  given  in  §  36,  (4).  Hence  we  have 


i  —  s, 
and 


REACTION  BETWEEN  SHIP  AND    PROPELLER. 


45.   AUGMENTATION  OF  RESISTANCE  DUE  TO  ACTION  OF 

PROPELLER. 

We  have  thus  far  considered  the  influence  of  the  ship, 
through  the  wake  which  it  produces,  on  the  action  of  the 
propeller.  We  now  turn  to  the  influence  of  the  propeller  on 
the  resistance  of  the  ship. 

We  have  already  in  §  37  considered  the  action  of  the  pro- 
peller in  producing  in  front  of  itself  a  defect  of  pressure  and 
thus  imparting  a  portion  of  the  total  acceleration  produced, 
before  the  water  acted  on  reaches  the  propeller  itself.  This 
defect  of  pressure  is  for  the  most  part  due  to  the  centrifugal 
force  consequent  upon  the  rotation  of  the  race  by  the  pro- 
peller. The  rotation  will  evidently  be  greater  as  the  blades 
stand  more  nearly  fore  and  aft,  and  hence  as  the  pitch-ratio  is 
higher.  This  defect  of  pressure  will  interfere  seriously  with 
the  natural  stream-line  motion.  The  water  instead  of  being 
able  to  close  around  the  stern  and  form  the  natural  stern  wave 
will  be  more  or  less  disturbed  and  drawn  away  to  the  pro- 
peller. The  result  of  this  is  a  diminution  of  the  pressure 
which  would  naturally  exist  about  the  stern  if  the  ship  were 
towed  at  the  same  speed.  The  result  of  this  is,  of  course, 
an  increase  in  the  amount  of  resistance  to  be  overcome  by 
the  propelling  agent.  This  increase  when  viewed  from  the 
standpoint  of  the  resistance  may  be  termed  the  augment  or 
augmentation  of  resistance. 

Where  paddle-wheels  are  used  as  the  propelling  agent,  a 
like  augmentation  is  experienced  though  it  arises  from  some- 
what different  causes.  The  action  of  the  paddle-wheels  at 
the  sides  and  nearly  amidships  gives  rise  to  a  race  of  water 
moving  faster  relative  to  the  ship  than  would  be  the  case  if 


23O  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

the  wheels  were  not  there.  The  result  is  an  increase  in  the 
skin-resistance,  The  wheels  also  undoubtedly  exercise  a 
disturbing  influence  on  the  natural  stream-line  motion,  and 
they  may  thus  introduce  an  additional  element  into  the 
augmentation  of  resistance.  The  amount  of  augmentation 
due  to  paddle-wheels  has  not  been  as  extensively  determined 
experimentally  as  for  screw  propellers,  but  various  experi- 
ments by  Dr.  Tidman,  R.  E.  Froude,  and  Messrs.  Denny 
indicate  that  it  does  not  much  differ  in  amount  from  that  for 
a  screw  propeller  under  like  conditions  of  speed. 

46.  ANALYSIS  OF  THE  POWER  NECESSARY  FOR  PROPULSION. 

Let  W=  the  indicated  horse-power; 

Wf  =  the  power  absorbed  by  the  friction  of  the  engine 

and  shafting,  and  by  any  attached  pumps. 
Then  W  —  Wf=  Wp  —  power  delivered  to  propeller. 

Let  7],  =  the  necessary  thrust,  supposing  the  ship  towed 

at  the  given  speed  z/; 

T=  the  actual  thrust  =  T0  -f-  the    amount  of    aug- 
mentation. 
Then  T0u  is  called  the  effective  horse-power;  and  Tu  is  called 

the  thrust  horse-power.  The  ratio  — ~  is  called  the  propul- 
sive coefficient. 

*TT* 

The  ratio  -~,  which  is  much  more  definitely  related  to 
Wp 

propulsive  efficiency,   has  unfortunately  received    no  special 
name.     We  shall  here  distinguish  it  as  the  coefficient  //. 
As  defined  in  §  33,  the  apparent  propeller  efficiency  is 

Tu 


REACTION  BETWEEN  SHIP  AND    PROPELLER.        231 

while  as  in  §  44  the  true  propeller  efficiency  is 


T(u  - 


u  — 


=  ct 


The  efficiency  ra  is  the  ultimate  test  of  the  performance  of 
the  propeller  considered  simply  as  a  propeller.  It  is  given  a 
certain  amount  of  work  Wp.  Working  in  undisturbed  water 
at  the  same  true  slip,  5a  =  (v  —  u  +  z>0),  it  would  develop  the 
thrust  T,  and  deliver  an  amount  of  work  T(v  —  5,)  = 
7\u  —  v0)  as  in  the  numerator  of  et  above. 

If,  however,  we  consider  the  propeller  simply  as  a  means 
of  getting  the  ship  through  the  water,  the  useful  work  will  be 
T0u,  since  this  is  the  amount  which  would  be  required  to  tow 
the  ship  at  the  speed  u.  The  ratio  T0u  -f-  Wp  is  therefore 
the  final  test  of  the  value  of  the  propeller  as  a  means  of 
actually  propelling  the  ship.  If  the  ship  produced  no  wake 
and  the  propeller  no  augmentation  of  resistance,  this  ratio 
7>  -^  Wp  and  the  ratio  e^  =  T(u  —  v0}  -±  Wp  would  be  the 
same.  The  relation  between  them,  therefore,  involves  the 
mutual  interaction  of  ship  and  propeller.  The  ratio  express- 
ing this  relation  is  hence  termed  the  hull  efficiency.  We 
have  therefore 

coefficient  h 


Hull  efficiency  = 


true  propeller  efficiency  ' 


or  coefficient  h  =  hull  efficiency  X  true  propeller  efficiency. 
Hence 

T0u       T(u  —  v0)       T0        u  T0. 

Hull  efficiency  =  TT7   ~- ^ •=-=.-       -  =  — (i  +2£/), 

\A/  II  /          U     -  —    *7f  / 

p  p  0 

and 


Coefficient  h  = 


/  u  — 


232  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

The  hull  efficiency  is  thus  seen  to  be  the  product  of  two 
factors  T0  -f-  T  and  u  -f-  (u  —  ^0).  The  first  is  the  ratio  of 
the  true  to  the  augmented  resistance,  or  the  reciprocal  of 
the  coefficient  of  augmentation  as  above  defined.  This  factor 
has  been  termed  by  R.  E.  Froude  the  "  Thrust-deduction 
factor."  We  shall,  however,  prefer  to  consider  it  as  the 
reciprocal  of  the  coefficient  of  augmentation,  T -T-  T0.  The 
effect  of  this  factor,  as  seen,  is  to  decrease  the  hull  efficiency 
in  the  ratio  7"0  ~-  T.  It  therefore  implies  a  loss  in  efficiency 
due  to  the  augmentation  of  resistance. 

The  other  factor  u  -=-  (u  —  v9)  —  (i  +  w),  as  already 
defined  in  §  44.  The  effect  of  this  factor,  as  seen,  is  to 
increase  the  hull  efficiency  in  the  ratio  (i  -\-  w).  It  therefore 
implies  a  gain  in  efficiency  due  to  the  location  of  the  propeller 
in  a  forward  wake  instead  of  in  undisturbed  water.  This  gain 
in  efficiency  is  seen  to  be  due  simply  to  the  gain  in  the  speed 
of  advance.  That  is,  if  the  propeller  were  working  in  undis- 
turbed water  at  the  same  revolutions  and  true  slip  5a  and 
consequent  thrust  T,  the  speed  of  advance  would  be  (v  —  52). 
In  the  actual  case,  due  to  the  fact  that  the  wake  water  is 
borne  forward,  so  to  speak,  to  meet  the  propeller,  the  same 
true  slip  5a  is  obtained  with  a  speed  of  advance  u  greater  than 
(v  —  52)  by  the  speed  of  the  wake  v9.  Hence  the  two  speeds 
of  advance  are  u  and  u  —  v9,  and  their  ratio  is  (i  -|-  w),  as 
defined. 

Of  the  two  factors  thus  constituting  hull  efficiency,  one  is 
seen  to  be  greater  and  the  other  less  than  I.  Hence  they 
tend  mutually  to  offset  each  other,  and  the  product  or  the 
hull  efficiency  will  usually  not  vary  widely  from  I.  In  fact 
many  determinations  by  R.  E.  Froude  indicate  that  the  hull 
efficiency  may  usually  be  taken  as  i  without  sensible  error, 


REACTION  BETWEEN  SHIP   AND   PROPELLER. 


233 


and  that  the  causes  which  seem  to  increase  one  factor  will 
correspondingly  decrease  the  other,  and  vice  versa.  Hence 
the  coefficient  //  may  usually  be  taken  as  sensibly  equal  to  the 
true  propeller  efficiency  r2.  It  should  not,  however,  be  for- 
gotten that  the  hull  efficiency  is  by  no  means  necessarily  I, 
and  that  circumstances  might  arise  in  which  such  an  assump- 
tion would  involve  a  sensible  error. 

This  analysis  of  the  total  power  and  the  various  relation- 
ships involved  may  be  illustrated  by  the  diagram  of  Fig.  66. 


FIG.  66. 

The   total   power,    revolutions,    and  thrust   are   supposed   to 

remain  constant.      The  subdivision  is  then  shown  both  with 

and  without  wake.     This  is  as  follows: 

AE  =  I.H.P.  =  total  power; 

AB  =  power  absorbed   by  friction   of   engine   and  attached 

pumps; 

BE  =  power  delivered  to  propeller  =  Wp\ 
BC  =  power  absorbed  in  the  eddies,  rotation,  and  sternward 

acceleration  communicated  to  the  water  acted  on, 

assuming  the  existence  of  a  wake; 
BCl  =  power  absorbed   under  the  preceding   head,  assuming 

that  there  is  no  wake  and  that  the  propeller  works 

in  undisturbed  water; 
CE  =  thrust  horse-power  =  7«,  assuming  the  existence  of  a 

wake ; 
C^E  =  thrust  horse-power  without  wake  =  T(u  —  z/0)  =  actual 

thrust  T  multiplied  by  the  reduced  speed  (u  —  vu) 

which    would    correspond    to    the    assumed    power, 

revolutions,  and  slip  with  no  wake; 


234  RESISTANCE  AND   PROPULSION   OF  SHIPS. 

FE  =  power  absorbed   by  the  augmentation   of  resistance. 

This  for  convenience  is  assumed  as  the  same  with  or 

without  wake; 
CF  =  power  absorbed  by  the  true  or  towed  resistance  in  the 

case  with  wake  =  effective  horse-power; 
Cf  =  power  absorbed  by  the  true  or  towed  resistance  in  the 

case  without  wake. 

Therefore  with  fixed  power,  revolutions,  thrust,  and  re- 
sistance the  wake  would  increase  the  speed  from  (u  —  v0)  to 
u  or  in  the  ratio  (i  +  w),  and  hence  the  useful  effect  in  the 
same  ratio. 

The  various  ratios  are  also  illustrated  as  follows: 

—r-=:  =  propulsive  coefficient ; 

CF 

=  coefficient  h; 


BE 
CE 


=  apparent  propeller  efficiency ; 


BE 

W  =  true 

CE 

-^-=  —  coefficient  of  augmentation ; 

Lr 

CE         u  u 

— -=  =  —  =  =  (i  -\-  w)  =  wake-return  factor; 


CF 

=  hull  efficiency. 


It  will  be  noted  that  the  diagram  of  Fig.  66  is  not  appli- 
cable to  the  same  ship  with  and  without  wake,  for  the  power, 
revolutions,  thrust,  and  resistance  are  supposed  to  remain 
constant,  while  the  speed  changes  from  (u  —  v0)  to  u.  The 


REACTION  BETWEEN  SHIP   AND    PROPELLER. 


235 


igram  is  applicable,  however,  to  the  performance  of  the 
propeller  with  and  without  wake,  under  the  conditions  that 
revolutions,  thrust,  and  resistance  shall  remain  constant,  and 
hence  that  the  propeller  at  the  two  speeds  (u  —  ^0)  and  u  shall 
be  opposed  to  the  same  resistance.  This  is  usually  the  most 
useful  mode  of  analysis,  as  it  serves  to  connect  the  propeller 
in  its  actual  surroundings  with  the  same  propeller  in  undis- 
turbed water,  developing  at  the  same  revolutions  and  true  slip 
the  same  thrust. 

We  may,  however,  as  in  Fig.  67,  illustrate  the  effect  of 
the  wake  on  the  same  ship  at  constant  speed,  resistance,  and 


AT 


c, 
FIG.  67. 


With 


Without 


thrust  of  the  propeller,  but  varying  revolutions  and  power. 
In  this  diagram  ABCFE  denote  for  the  ship  with  wake  the 
same  points  as  in  Fig.  66,  while  A^B^Cf^E^  denote  similar 
points  for  the  same  ship  at  the  same  speed  and  resistance, 
but  without  wake.  The  power  represented  by  CE  is  of 
course  equal  to  that  represented  by  CJE^  The  propeller 
power  BE  and  the  total  power  AE  are,  however,  less  than 
the  corresponding  amounts  BlEl  and  A^E^  This  saving  of 
power  comes  about  as  follows:  As  shown  in  §  36,  (i),  thrust 
varies  with  revolutions  and  with  slip.  Now  if  there  were  no 
wake  and  the  slip  were  St,  the  necessary  thrust  would  be 
developed  by  a  certain  number  of  revolutions.  With  the 
wake  v9  and  the  same  speed  u  the  slip  becomes  increased  to 
5,  +  ?;0  —  -$,»  and  the  revolutions  necessary  to  develop  the 
fixed  thrust  are  correspondingly  decreased.  Now  with  con- 
stant thrust  the  turning  moment  or  torque,  and  hence  the 


236  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

mean  effective  pressure  in  the  cylinders,  will  all  remain  very 
nearly  constant.  Hence  the  propeller  horse-power  Wp  and 
the  indicated  horse-power  W  will  vary  very  nearly  as  the 
revolutions.  Hence  while  the  work  CE  will  equal  £",-£",,  the 
work  BE  and  the  total  work  AE  will  be  less  respectively  than 
BlEl  and  A^E^  very  nearly  in  the  same  ratio  as  that  in  which 
the  revolutions  are  reduced.  In  this  way,  then,  it  is  seen  that 
the  presence  of  the  wake  makes  it  possible  to  obtain  the 
necessary  thrust  and  propulsion  of  the  ship  with  a  lower 
number  of  revolutions  and  with  a  correspondingly  decreased 
engine-power  than  in  undisturbed  water. 

The  wake  is  due  to  the  motion  of  the  ship,  and  its  kinetic 
energy  is  simply  the  energy  which  has  been  put  into  it  by  the 
ship  in  its  movement  through  the  water.  The  energy  of  the 
wake  comes  therefore  from  the  engine,  or  more  ultimately 
from  the  coal  or  fuel,  as  the  source  of  energy.  The  reduc- 
tion of  the  engine-power  and  fuel-consumption,  as  above 
explained,  may  therefore  be  considered  as  a  return  from  the 
wake,  or  as  a  reutilization  of  a  small  part  of  the  energy  which 
has  been  expended  in  its  formation. 

The  amount  of  this  reduction  will  depend  on  the  relation 
of  thrust,  revolutions,  and  slip,  and  cannot  be  expressed  in 
simple  terms.  From  §  36,  (14)  and  (16),  however,  we  have 


=  aPProximately  ^'(-°34  +  .85*), 


pN(i  -s) 


where  B  includes  all  terms  relating  to  the  geometry  of  the 
propeller.      Hence 

T 

N*    ~-  £(.034  +  .855) (I) 


REACTION  BETWEEN  SHIP  AND    PROPELLER.        237 


The  effect  of  z>0  on  s  is  readily  determined,  and  for  con- 
stant T  the  influence  of  this  on  N*  is  shown  approximately 
by  this  equation.  Hence  follows  the  effect  on  N  and  on 
power.  As  an  illustration,  suppose  j,  =  .15  and  w  =  .133. 
Then  from  §  44,  (4),  s9  =  .25,  and  from  (i)  above 


.2465 
.1615 


=  1.526     and 


=   1.24, 


Hence  with  these  values  the  propulsion  without  the  wake 
would  require  the  development  of  about  24  per  cent  more 
power  than  in  the  actual  case. 

We  will  now  give  a  numerical  illustration  of  the  analysis 
illustrated  in  Fig.  66. 

Let  st  =  .16; 

*,  =  -27. 

This  would  mean  that  the  propeller  in  undisturbed  water 
with  27  per  cent  slip  would  give  the  same  thrust  as  obtained 
in  the  actual  case  at  16  per  cent  apparent  slip,  the  revolutions 
being  the  same  in  each  case. 

Then  from  §44,  1  +  ^=1.151; 

w  =    .151. 


Let  AE,  the  I.H.P.,  be  denoted  by 
Let  AD  be 


Let  CE  =  Tu  .............  .  . 

Then  el  =  apparent  propeller  efficiency 
And  et  =  true  propeller  efficiency 
Also  dE=CE+  i  +  w 
Hence  CC. 


14 
.86 
.67 
.779 
.677 
.582 
.088 


238  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

Let  coefficient  of  augmentation  be =  1.165 

Then  CF  =  E.H.P =  .575 

And  FE =  .095 

Then  coefficient  //..... =  .669 

Propulsive  coefficient =  .575 

Hull  efficiency ==  .988 

The  following  table  shows  the  results  of  certain  experi- 
ments carried  out  on  a  Dutch  tugboat.*  During  the  trials 
the  thrusts  were  measured  at  the  thrust-block  by  a  hydraulic 
dynamometer,  while  the  values  of  the  E.H.P.  were  deter- 
mined from  model  experiments. 


I.H.P. 

T.H.P. 

E.H.P. 

Rev. 

Speed. 

Principal  Dimensions. 

3I-03 

19.76 

15-80 

94 

6.97 

50.56 

33.16 

27.42 

in 

8.07 

Length                =  72' 

80.24 

53-22 

42-74 

127-5 

9.02 

Beam                   =  14'      9" 

132.35 

89-43 

70.69 

148 

10.07 

Draft  forward  =    3'    10" 

170-83 

118.85 

87.75 

160.5 

10.47 

Draft  aft            =     7'    4!" 

230.58 

161  .40 

108.46 

175 

10.84 

Displacement   =  69  tons 

260.32 

180.29 

I  2O.  22 

180.5 

II.  OI 

The  ratio  of  T.H.P.  to  I.H.P.  varies  with  increase  of 
speed  from  .64  to  .69,  and  that  of  E.H.P.  to  I.H.P.  from 
.54  to  .46.  The  pitch  of  the  propeller  was  7.63  feet,  and, 
as  may  be  determined,  the  slip  varied  from  about  1.5  to  19 
per  cent.  The  increase  of  propulsive  coefficient  at  low 
values  of  the  speed  and  slip  is  somewhat  abnormal,  but 
whether  due  to  errors  in  the  data  or  to  an  increased  value 
of  the  ship  efficiency  at  these  points  does  not  appear.  The 
series  of  experiments  is  a  valuable  one,  and  it  may  be  hoped 
that  the  near  future  will  give  us  many  more  of  the  same 
character. 

*  See  Steamship  for  October  1897. 


REACTION  BETWEEN  SHIP  AND    PROPELLER.        239 


47.  INDICATED  THRUST. 

This  term,  which  frequently  occurs  in  engineering  litera- 
ture, is  defined  as  follows: 


33000  I.H.P. 


Indicated  thrust  in  tons  = 


This  is  a  thrust  which  would  correspond  to  an  absence  of 
slip  and  a  total  efficiency  of  i.  This  is  a  wholly  impossible 
set  of  conditions,  but  as  it  is  considered  convenient  for  the 
expression  of  certain  relationships,  we  will  show  its  connection 
with  a  much  better  known  quantity.  Let  C  be  an  engine 
constant  defined  by  the  equation 


C  = 


L.P.  cylinder  area  X  2  X  stroke  in  feet 
33000 


33000 


Then  the  mean  effective  pressure  reduced  to  the  L.P.  cylin- 
der is  the  pressure  defined  by  the  equation 


, v 

(m.e.p.)  = 


33000  I.H.P. 


(3) 


Now  comparing  indicated  thrust  and  reduced  mean  effec- 
tive pressure,  it  is  readily  seen  that  there  is  a  constant  ratio 
between  them,  and  hence  that  the  one  is  in  constant  pro- 
portion to  the  other.  Hence  we  may  remember  that  the 
indicated  thrust  is  simply  a  quantity  proportional  to  the 
reduced  mean  effective  pressure. 

We  will  also  show  here  another  expression  for  the  re- 
duced mean  effective  pressure.  Let  Alt  A»  At  be  the  areas 
of  the  successive  cylinders  of  a  triple-expansion  engine,  A, 
being  the  value  for  the  low-pressure  cylinder.  Let  /t,  p^ 


24O  RESISTANCE  AND   PROPULSION    OF  SHIPS. 

and  /,  be  the  successive  actual  mean  effective  pressures  in 
the  same  cylinders.     Then  evidently 


(4) 


The  values  in  (3)  and  (4)  are  evidently  the  same.  For 
multiple-expansion  engines  of  any  number  of  stages  the 
same  general  method  applies.  For  triple-expansion  engines 
with  initial  pressures  from  160  to  1  80  pounds  absolute,  or  145 
to  165  by  gauge,  the  reduced  mean  effective  pressure  is 
usually  from  30  to  40  pounds  per  square  inch. 

48.  NEGATIVE  APPARENT  SLIP. 

If  there  were  no  wake,  the  apparent  slip  sl  and  the  true 
slip  s9  would  be  the  same.  With  the  development  of  the 
wake,  however,  the  true  slip  remaining  the  same,  the  appa- 
rent slip  decreases  until  some  point  is  reached  where  an 
equilibrium  of  conditions  is  maintained. 

From  §  44  we  have 

sl  =  s9(i  +  w)  —  w  =  s>  —  (i  —  ss)w. 

Now  if,  as  stated  above,  s9  remains  constant  and  w 
increases,  sl  will  continuously  decrease.  If  w  reaches  a  suffi- 
cient value,  5,  will  become  o  and  then  negative.  The  condi- 
tion that  st  =  o  is 


or     s,  =       —*  •     •     (0 


REACTION  BETWEEN  SHIP   AND    PROPELLER. 


24 1 


The  question  of  the  possibility  or  otherwise  of  a  o  or 
negative  value  of  s1  is  simply  a  question  of  the  possibility  or 
otherwise  of  a  wake  value  equal  to  or  greater  than  this  value 
j,  -T-  (i  —  s^).  There  is  no  inherent  reason  why  such  a  wake 
should  not  exist,  or  why  it  should  not  be  formed  by  the 
motion  of  the  ship  through  the  water. 

Such  a  condition  is  not,  however,  desirable,  or  indicative 
of  a  good  propeller  efficiency.  It  indicates  the  existence  of 
either  one  or  both  of  the  following: 

(1)  A  low  true  slip,  and  consequently  a  low  efficiency  as 
shown  in  all  efficiency  curves,  as  in  §  36. 

(2)  An    excessive    ship-resistance    corresponding    to    the 
formation  of  a  wake  of  this  velocity. 

While,  therefore,  the  apparent  propeller  efficiency  will  be 
high,  the  true  efficiency  may  nevertheless  be  quite  low,  and 
the  actual  resistance  per  ton  of  displacement  will  naturally  be 
excessive. 

As  a  slightly  different  way  of  expressing  the  condition  of 
o  or  negative  slip,  we  may  take  §  42,  (2).  Thus 

5,  =  5,  -  v.. 
Hence  if  5,  <  o,   Sa  <  v9. 

This  may  be  considered  as  implying  simply  that  the  pro- 
peller is  capable  of  furnishing  the  necessary  thrust  with  a  true 
slip  5,  equal   to  or  less  than  the  wake-velocity  v0.     This  i&- 
evidently  equivalent  to  the  condition  expressed  in  (i) 

_       w 


Now  in  Chapter  IV  practical  methods  will  be  given  for 
designing  a  propeller  to  fulfil  any  given  program  of  conditions. 
To  design  a  propeller  which  wo-;  Id  probably  show  o  or  nega- 


242  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

tive  slip  we  have  therefore  simply  to  fix  as  one  of  the  condi- 
tions a  value  of  s^  equal  to  or  less  than  that  given  by  (i), 
using  such  value  of  w  as  shall  be  deemed  appropriate.  Thus 
for  illustration,  the  value  of  w  is  very  commonly  about  .  i. 
Hence  w  H-  (i  -f-  w)  —  .09,  and  a  propeller  designed  to  fulfil 
the  given  conditions  with  a  true  slip  of  from  7  to  9  per  cent 
would  probably  show  a  o  or  slightly  negative  apparent  slip. 
Such  a  propeller  would  be  very  large  and  wasteful,  and,  as 
above  stated,  quite  undesirable.  These  results  in  actual 
practice  are  avoided  by  the  assumption  of  a  much  larger 
value  of  the  true  slip.  Their  mention  is  simply  introduced 
here  in  order  to  show  the  entire  possibility  and  significance  of 
a  o  or  negative  value  of  the  apparent  slip.  While,  therefore, 
the  possibility  of  these  results  is  unquestionable,  it  is,  how- 
ever, presumable  that  many  reported  cases  of  negative  slip 
have  arisen  from  errors  of  measurement,  especially  in  the 
pitch.  In  particular  might  this  be  the  case  with  propellers  of 
variable  pitch,  in  which  the  definition  of  mean  pitch  is  neces- 
sarily arbitrary  in  character,  according  as  we  make  it  purely 
geometrical,  or  give  it  a  dynamical  basis  by  giving  to  the 
pitch  of  each  element  a  weight  proportional  to  the  thrust 
which  it  develops  or  the  work  which  it  absorbs.  Due  to  this 
necessarily  arbitrary  character  of  the  definition  of  mean  pitch 
with  such  propellers,  the  expressions  mean  pitch  and  apparent 
slip  lose  much  of  the  significance  which  we  are  able  to  give 
to  them  with  propellers  of  uniform  pitch.  See  also  §  42. 

The  distribution  of  forces  on  a  screw-propeller  blade,  re- 
ferred to  in  §§  42  and  52,  Figs.  64  and  85,  indicates  further- 
more the  possibility  of  a  small  positive  thrust  with  a  small 
negative  true  slip  measured  with  reference  to  the  face.  The 
slip  of  the  water  (§  42),  however,  must  always  be  positive. 


CHAPTER    IV. 
PROPELLER   DESIGN. 

49.  CONNECTION  OF  MODEL  EXPERIMENTS  WITH  ACTUAL 

PROPELLERS. 

WE  now  take  up  the  considerations  relating  to  the  appli- 
cation of  experimental  data  derived  from  models  to  the  design 
of  full-sized  propellers. 

We  must  first  remember  that  the  actual  data  given  by 
Froude's  experiments,  as  described  in  §  36,  relate  to  certain 
propellers  .68  ft.  in  diameter,  with  certain  pitch-ratios,  at  cer- 
tain revolutions  and  slips,  and  with  blades  of  an  elliptical  shape, 
of  a  certain  area,  material,  and  thickness,  and  with  hubs  of 
a  certain  diameter  relative  to  that  of  the  propeller  itself. 
The  data  thus  found  was  so  regular  in  character  and  agreed  so 
well  with  the  fundamental  propositions  (a),  (b),  §  36,  that  it 
seemed  fair  to  accept  these  propositions  as  empirically  true 
for  the  range  of  values  covered,  and  hence  to  accept  as  re- 
liable the  interpolations  thereby  effected.  It  therefore  follows 
that  the  results  may  be  considered  as  applicable  to  all  pro- 
pellers of  this  diameter  and  character  of  blade  within  the 
given  limits  of  pitch-ratio,  and  working  between  the  given 
limits  of  slip.  We  will  now  consider  the  justice  of  extend- 
ing these  results  to  full-sized  propellers. 

The  derivation  of  §  36,  (11),  (12),  shows  that  the  whole 
question  depends  on  the  supposition  that  the  forces  P  and  Q 

243 


244  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

vary  as  the  area  and  as  the  square  of  the  speed,  or  that  the 
values  of  A",  L,  and  K  -f-  L  =  e  are  independent  of  actual 
dimensions,  and  are  simply  functions  of  pitch-ratio  and  slip. 

In  regard  to  efficiency,  a  satisfactory  general  correspond- 
ence between  the  values  for  model  experiments  and  actual 
propellers  has  been  testified  to  by  numerous  comparisons.  In 
particular  it  has  been  found  that  the  general  shape  of  the 
efficiency-curve,  as  shown  in  Fig.  60,  is  characteristic  of  the 
performance  of  full-sized  propellers.  According  to  Mr. 
Froude's  comparisons,  we  may  depend  safely  on  the  general 
character  of  the  efficiency  data  and  on  the  values  themselves 
considered  relatively,  while  absolutely  they  may  involve  a 
slight  error.  That  is,  we  may  safely  depend  on  the  experi- 
mental data  to  show  us  correctly  the  general  conditions  for 
the  best  efficiency  and  how  the  efficiency  will  vary  for  given 
changes  in  pitch-ratio  and  slip,  while  there  may  be  slight 
errors  in  the  actual  values  of  the  efficiency  itself.  In  any  case 
the  amount  of  error  is  presumably  small,  and  experience  seems 
to  indicate  that  we  shall  be  quite  safe  in  using  these  data,  at 
least  as  a  general  guide  to  the  desirable  range  of  values  of 
pitch-ratio  and  slip,  within  which  to  work  in  any  given  case  of 
design. 

The  changes  which  may  be  involved  in  passing  from  the 
model  to  the  full-sized  propeller  are  of  three  kinds : 

(1)  Change  in  diameter,  and  hence  in  all  other  dimensions 
in  proportion ; 

nd* 

(2)  Change  in  area- ratio,  h  =  A  -. ; 

4 

(3)  Change  in  shape  of  blade. 

In  any  actual  case  any  or  all  of  these  might  be  involved. 
The  above  remarks  relating  to  efficiency  imply  only  change 


PROPELLER   DESIGN. 


245 


i).  That  is,  geometrical  similarity  between  the  model  and  the 
propeller  is  supposed  to  be  maintained.  Hence  only  under 
this  condition  are  the  results  of  the  model  experiments  strictly 
applicable,  and  as  variations  (2)  and  (3)  enter  in  to  a  greater 
and  greater  degree,  the  use  of  these  results  must  be  attended 
with  a  continually  decreasing  degree  of  accuracy  and  confi- 
dence. It  would  be,  however,  a  great  convenience  if  we 
might  feel  the  liberty  of  introducing  within  moderate  limits 
changes  (2)  and  (3),  or  more  especially  the  former.  Now 
mathematical  investigation,  into  the  details  of  which  we  will 
not  enter  here,  indicates  that  the  variation  of  efficiency  with 
variations  (2)  and  (3)  is  very  slow,  and  hence  that  we  may 
presumably  introduce  such  changes  in  moderate  degree  with- 
out sensibly  affecting  the  efficiency. 

Turning  now  to  the  value  of  AT  in  §  36,  (i  i),  we  note  that 
it  involves  geometrical  ratios  and  the  two  coefficients  a  and  f. 
Let  us  now  consider  the  effect  of  changes  (i),  (2),  and  (3)  on 
these  coefficients. 

From  the  close  analogy  between /and  the  skin-resistance 
coefficient  of  §  7,  it  would  seem  probable  that  it  would  be 
more  especially  affected  by  change  (i),  and  hence  that  it 
should  receive  some  correction  similar  to  that  for  length,  as 
discussed  in  that  section.  The  data  for  such  correction  do 
not  exist,  but  such  indications  as  are  available  indicate  that  it 
is  not  large  in  amount.  Of  still  less  importance  are  the  effects 
due  to  (2)  and  (3). 

For  the  coefficient  a  the  case  is  by  no  means  the  same. 
Considering,  as  we  fairly  may,  that  so  long  as  proportion  and 
form  remain  the  same  the  total  normal  resistance  varies  as  the 
area  and  as  the  square  of  the  speed,  it  follows  that  a  may  be 
considered  independent  of  change  (i).  Its  variation  with 


246  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

change  (3)  within  the  limits  usually  involved  will  also  be  slight, 
We  come,  therefore,  to  the  variation  of  a  with  area-ratio  hr 
all  other  characteristics  remaining  the  same. 

Suppose  we  start  with  very  narrow,  thin,  elliptical  blades 
of  maximum  width  .01  ir,  or  area-ratio  h  =  .01.  These  blades 
with  given  revolutions  and  slip  will  develop  a  certain  thrust 
T  and  useful  work  U  corresponding  to  certain  average  and 
distributed  values  of  the  coefficients  a  and/.  Now  if  the 
width  and  hence  the  area  of  these  blades  be  doubled,  the  area- 
ratio  being  now  .02,  we  shall  undoubtedly  find  that  the  value 
of  a  will  remain  sensibly  unchanged,  and  hence  the  total 
thrust  will  be  doubled.  If  we  continue  thus  to  increase  the 
width,  and  hence  the  area,  we  shall  undoubtedly  for  a  time  find 
the  same  rate  of  increase  in  the  thrust,  indicating  a  sensibly 
constant  value  of  a.  If  the  path  of  the  blade  were  rectilinear, 
this  would  undoubtedly  hold  true  up  to  some  increase  of  area 
beyond  actual  experience.  With  the  screw  propeller,  however, 
the  path  is  helicoidal;  and  we  readily  see  that  as  the  width 
and  area  are  increased  the  thrust  cannot  increase  indefinitely, 
but  must  rather  tend  toward  a  limit.  In  other  words,  as  the 
area  is  increased  the  thrust  is  increased  at  a  slower  and  slower 
rate.  This  implies  a  gradual  and  continual  decrease  in  the  co- 
efficient a,  while  presumably  the  value  of  f  remains  nearly  the 
same,  or  at  least  falls  off  much  less  rapidly  than  a.  This  de- 
crease in  a  is  due  to  mutual  interference  in  the  streams  acted  on 
by  the  different  blades.  While  therefore  a  decreases,  f  re- 
mains nearly  the  same,  and  the  thrust  per  unit  area  falls  off 
accordingly.  In  this  way  we  shall  finally  reach  a  point  where 
increase  of  area  gives  no  increase  of  thrust.  The  value  thus 
developed  is  therefore  a  maximum,  and  cannot  be  exceeded, 
no  matter  what  the  area.  Indeed,  certain  indications  seem. 


PROPELLER   DESIGN. 


247 


to  point  toward  a  possible  falling  off  of  thrust  with  excessive 
increase  of  area. 

The  ideal  maximum  thrust  obtainable  for  given  diameter, 
pitch,  revolutions,  and  slip  is  that  corresponding  to  the  for- 
mation of  what  is  termed  a  complete  column  ;  that  is,  a  col- 
umn of  water  in  which  each  part  has  received  the  full  acceler- 
ation which,  with  geometrically  perfect  action,  the  propeller 
is  capable  of  imparting.  The  amount  of  this  maximum  thrust 
may  be  investigated  as  follows: 

Referring  to  Fig.  55,  it  is  shown  that  the  value  of  the 
longitudinal  acceleration  is  given  by 

EG  =  EC  cosa  a  =  Si  cos2  a  =  ^/TVcos3  a. 

Now  for  an  element  of  the  total  column  consisting  of  a 
shell  of  water  of  thickness  dr,  moving  with  this  acceleration, 
the  thrust  will  be 

dT  =  mass  of  water  per  second  X  acceleration  EG. 
Hence 

dT  =  -nd  .  dr  .  velocity  of  feed  X  EG. 

o 

We  consider  the  velocity  of  feed  as  represented  by  DG, 
or  as  the  speed  of  advance  DE  -f-  the  acceleration  EG.  The 
value  of  EG  above  and  found  in  §  42  assumes  the  angle  CAE 
small.  On  the  same  assumption  we  may  also  put 

CG  =  EC  sin3  a  =  S9  sin*  a  =  s^Nsin*  a. 
Hence 

DG  —  pN(\  —  J9sinaa); 

and  putting  in  general  s  for  s9,  we  have 


—  -nd  .  drsp*N*  cos'  a(i  —  s  sin«  a). 

o 


248  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

But  tan  a  —  p  -=-  nd,    and    d  =  2r  =  py.      Making    these 
substitutions  and  reducing,  we  find 


(0 


Hence  for  the  entire  propeller  the  maximum  ideal  thrust 
will  be  given  by  the  integration  of  this  expression  between 
the  proper  limits  for  y.  This  is  seen  to  be  entirely  independ- 
ent of  the  breadth  of  blade  or  area  of  its  surface.  The  thrust 
thus  found  would  be  the  maximum  possible  on  the  supposi- 
tions made,  and  would  correspond  to  the  perfect  action  of 
each  element,  and  to  the  absence  of  skin-resistance.  It  is 
therefore  greater  than  could  be  obtained  in  any  actual  case,  no 
matter  what  the  amount  of  surface. 

On  the  assumption  that  the  value  of  a  is  constant,  Cot- 
terill*  has  used  the  above  expression  for  dT  to  find  the 
breadth  of  blade  necessary  to  form  a  complete  column  at 
various  values  ofjj/.  This  was  done  by  equating  the  thrust 
above  to  that  in  a  form  similar  to  §  35,  (6),  omitting  /and 
with  an  assumed  value  1.7  for  a,  and  solving  for  dA.  Since, 
however,  this  value  is  not  accurate  for  a,  and  since,  more- 
over, it  cannot  remain  constant  but  must  fall  off  in  marked 
degree  with  large  increase  of  area,  the  results  thus  found  can 
only  be  considered  as  rough  approximations  to  the  lower 
limits,  beyond  which  we  must  go  in  order  to  approach  the 
41  complete  column  "  condition.  So  far  as  these  indications 
went,  however,  they  showed  that  propellers  of  ordinary  pro- 
portions do  not  form  complete  columns,  and  hence  do  not 
give  the  maximum  thrust  corresponding  to  their  diameter, 

*  Transactions  Institute  of  Naval  Architects,  vol.  xx.  p.  152. 


PROPELLER   DESIGN. 


249 


ch,  revolutions,  and  slip.  This  instead  of  being  a  fault  is 
a  decided  advantage,  for  it  is  very  sure  that  a  propeller  work- 
ing near  the  upper  limit  of  the  thrust  would  be  less  efficient 
than  if  the  surface  and  thrust  were  less,  all  other  conditions 
remaining  the  same.  This  arises  from  the  fact  that  the  last 
increments  of  thrust  must  be  obtained  by  the  addition  of  a 
disproportionate  amount  of  surface  with  its  accompanying 
skin-resistance.  This  will  result  in  an  increase  of /relative  to 
a,  and  in  a  corresponding  loss  in  efficiency  (§  36,  (4)).  Hence 
while  such  a  propeller  may  give  a  large  thrust  for  its  diame- 
ter, pitch,  slip,  and  revolutions,  the  proportion  of  useful  to 
total  work  will  be  comparatively  poor. 

Let  us  now  return  to  the  expression  for  the  thrust  in  (i). 
By  integrating  the  function  of  y  between  various  values  of  the 
outer  limit  we  obtain  the  values  of  the  maximum  ideal  thrust 
for  the  entire  propellers  of  corresponding  pitch-ratios.  A 
comparison  of  these  with  the  values  actually  obtained  by  ex- 
periment under  similar  conditions  of  diameter,  pitch,  revolu- 
tions, and  slips  will  be  of  interest. 

The  integrations  were  effected  by  approximate  methods, 
and  the  results  are  shown  by  CD,  Fig.  68.  These  are  plotted 
to  represent  the  maximum  ideal  thrust  in  tons  for  propellers 
of  10  feet  diameter  at  100  revolutions  and  20  per  cent  slip, 
and  of  various  values  of  the  pitch-ratio  as  given  on  the  axis  of 
abscissae.  The  curve  AB'm  the  same  diagram  gives  similarly 
the  actual  values  of  the  thrust  as  derived  for  the  same  condi- 
tions from  Froude's  experimental  models,  assuming  geomet- 
rical similarity.  This  curve  gives  therefore  the  actual  thrusts 
for  propellers  the  same  as  above,  and  of  area-ratio  h  =  .36. 
With  other  values  of  the  slip  the  results  were  entirely  similar, 
thus  indicating  still  more  clearly  that  the  propeller  of  usual 


250 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


proportions  is  far  from  developing  the  maximum  ideal  thrust. 
The  ratio  of  these  two  thrusts  is  shown  by  the  curve  EF. 


u 


1.2         1.3 


1.4        1.5        1.6 

PITCH  RATIO 

FIG.  68. 


1.7 


1.8        1.9        2.0 


It  would  be  more  especially  useful  to  know  the  law  con- 
necting increase  of  thrust  with  h.  Unfortunately  the  infor- 
mation on  this  point  is  very  scanty.  R.  E.  Froude  gives  as 


PROPELLER   DESIGN. 


251 


the  relative  thrusts  of  4,  3,  and  2   blades,  each  of  the  same 
form  and  area  as  in  Fig.  57,  the  numbers 

i:  .865:  .65, 
while  the  areas  are  in  the  ratios 

i:  .75:  50. 

This  shows  a  gain  in  the  thrust  per  unit  area  as  the  area 
is  decreased.  It  would  seem  fair  to  assume  this  gain  to  be 
due  fundamentally  to  the  decrease  in  area.  This  is  also  in 
accord  with  Isherwood's  experiments,*  which  indicated  that 
for  a  given  blade  area  the  thrust  was  practically  independent 
of  the  number  of  blades.  We  will  assume  therefore  that  the 
above  relative  thrusts  and  areas  correspond,  the  area  i,  how- 
ever, corresponding  to  an  area-ratio  of  .36  and  the  others 
respectively  to  .27  and  .18.  The  relative  values  of  the 
thrusts  will  certainly  depend  on  the  pitch-ratio  and  on  the 
slip,  so  that  those  given  above  cannot  be  true  generally. 
They  are  believed  to  apply  more  particularly  to  a  value  of 
the  pitch-ratio  about  1.3,  and  to  average  values  of  the  slip. 
For  propellers  of  about  this  pitch-ratio  we  have  therefore 
three  values  as  above,  a  zero-point  for  h  =  o,  and  an  ideal 
maximum  as  in  Fig.  68.  Let  us  now  take  arbitrarily  the 
thrust  of  the  four-bladed  propeller — or  rather  the  thrust  for 
area-ratio  .36 — as  i,  and  indicate  our  results  graphically. 
This  is  shown  in  Fig.  69,  where  the  abscissae  give  the  values 
of  //  and  the  ordinates  the  values  of  the  thrust  in  terms  of 
that  for  //  =  .36,  as  unity.  P,  Q,  and  R  are  the  points  for  2, 
3,  and  4  blades,  or  for  //  =  .18,  .27,  .36,  the  corresponding 

*  Engineering,  vol.  xx.  pp.  369,  370. 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 

ordinates  being  as  plotted,  .65,  .865,  I.  In  Fig.  68  the  rela- 
tive value  of  the  ideal  maximum  for  pitch-ratio  1.3  is  seen  to 
be  1.3.  This  value  is  laid  off  as  the  line  CD.  We  have, 
therefore,  <9,  P,  Q,  and  R  as  experimental  points,  with  CD  as 
an  ideal  limit,  the  actual  limit  being  slightly  less.  The  con- 
clusion seems  clear  that  the  variation  of  thrust  with  h  must 
follow  some  such  line  as  OPQRS. 

For  any  other  pitch-ratio  a  similar  curve  must  pass 
through  R,  since  we  take  the  thrust  for  h  .—  .36  as  rela- 
tively i.  Now  putting  in  the  upper  limits  for  pitch-ratio  —  2 
and  I,  we  have  AB  and  EF.  While  the  data  on  which  we 
are  working  is  very  meagre,  it  seems  to  be  a  fair  conclusion 
that  the  curves  for  these  values  of  the  pitch-ratio  will  be 
similar  to  OHV  and  OGT,  with  intermediate  curves  for 
intermediate  values  of  the  pitch-ratio.* 

Mention  may  also  be  made  here  of  the  experiments  of 
Mr.  A.  Blechynden,t  the  results  of  which  indicate  a  general 
-confirmation  of  the  conclusions  we  have  just  drawn. 

From  the  diagrams  of  Fig.  69  it  would  follow  that  with 
a  propeller  of  high  pitch-ratio  the  thrust  increases  nearly  as 
the  area  to  a  considerably  greater  value  of  h  than  with  low 
pitch-ratio.  Also  remembering  that  with  present  practice  h 
is  usually  found  between  .35  and  .45,  it  is  seen  that  a  rela- 
tive increase  of  area  is  much  more  admissible  on  a  propeller 
of  high  pitch-ratio  than  on  one  of  low,  and  that  in  the  latter 

*  An  experimental  investigation  of  the  influence  of  the  amount  of  sur- 
face on  the  performance  of  screw  propellers  is  now  being  carried  on  by  the 
author,  the  results  of  which,  so  far  as  determined,  justify  these  general 
conclusions,  drawn  independently  of  the  experimental  work.  For  a  pre- 
liminary statement  of  the  work,  reference  may  be  made  to  the  Transac- 
tions of  the  Society  of  Naval  Architects  and  Marine  Engineers,  vol.  v. 

f  Transactions  N.  E.  Coast  Institute  of  Engineers  and  Ship-builders, 
vol.  in.  p.  179. 


PROPELLER   DESIGN. 


253 


case  we  are  near  the  point  where  increase  of  area  will  give 
very  slight  return  in  increase  of  thrust.  These  conclusions 
also  are  borne  out  by  the  indications  of  actual  experience, 
and  should  be  borne  in  mind  in  connection  with  the  proper 
value  of  /i. 

It   must  be  remembered  that  there  is  very  little  experi- 


i.s 


l.o 


1.4 


Sl.2 


e  1.0 

o 

o 

»- 
<    .8 


S    .6 


2.0 


.3 


.4  .5  .6 

SURFACE  RATIO 
FlG.    69. 


.9 


1.0 


mental  data  bearing  on  the  relations  shown  in  Fig.  69,  and 
the  diagram  is  simply  an  attempt  to  express  the  available 
data,  extended  by  reference  to  an  ideal  maximum  which  may 
be  computed  as  a  function  of  the  conditions  of  operation  of 
the  propeller. 

We  have  thus  shown  the  general   nature  of  the  variation 


254  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

of  thrust,  useful  work,  and  hence  of  the  coefficient  a,  with 
variation  in  h. 

Now  of  the  three  kinds  of  change  with  which  we  may  be 
involved  in  passing  from  the  model  to  the  full-sized  propeller, 
let  us  first  consider  only  (i)  and  (2).  From  the  preceding 
discussion  it  will  be  seen  that  the  change  of  a  with  //  and  its 
resultant  effect  on  thrust  and  useful  work  is  presumably  the 
only  one  of  importance,  and  the  only  one  for  which  we  shall 
need  to  provide.  Referring  to  §  36,  (10)  and  (11),  it  is  seen 
that  this  variation  is  equivalent  to  a  change  in  the  value  of 
the  integrals  H  or  K,  the  values  of  which  should  gradually 
decrease  as  h  increases,  thus  corresponding  to  the  decreasing 
rate  of  increase  of  thrust  with  area.  Instead  of  attempting 
to  express  this  variation  in  K  as  a  function  of  pitch-ratio  and 
h,  let  us  simply  represent  its  ratio  to  the  standard  value,  or 
value  for  h  =  .36,  by  a  factor  m.  We  have  then,  instead  of 

§  36,  (H), 

U=(p'NJd'*klm (2) 

The  value  of  m  will  depend  on  h  and  pitch-ratio.  If 
h  —  .36  then  m  is  always  I,  and  (2)  becomes  the  same  as 
§  36,  (14),  corresponding  thus  to  the  general  condition  of 
geometrical  similarity  between  the  model  and  the  propeller. 
If  h  is  not  .36,  then  a  value  for  m  must  be  selected  having  in 
view  the  general  results  indicated  in  Fig.  69.  It  is  evident, 
in  fact,  that  the  values  of  m  are  represented  by  the  ordinates 
to  the  curves  OGT,  OPS,  etc.  Hence  by  inspection  a  value 
may  be  selected  which  shall  approximately  correspond  to  the 
given  conditions  of  pitch-ratio  and  h. 

It  should  be  especially  noted  that  as  here  treated  the 
question  of  the  number  of  blades  does  not  directly  enter. 
The  value  of  m  is  determined  by  the  area  and  not  by  the 


PROPELLER   DESIGN. 


255 


number  of  blades.  Some  attention  must,  of  course,  be  paid 
to  the  usual  proportions  between  length  and  breadth  of 
blade,  but  area  is  made  the  fundamentally  controlling  feature. 
In  general  we  shall  find  the  following  a  safe  guide: 

With  area-ratio  from  .15  to  .25,  2  blades  may  be  used; 
"     .25  to  .40,  3       " 
"     .35  to  .50,  4       " 

We  may  now  consider  the  effect  due  to  change  (3) — a 
change  in  the  form  of  the  blade.  The  effect  arises  here  not 
from  a  change  in  a  or/,  but  from  a  change  in  the  form  of  dA 
as  a  function  of  y. 

The  thrust  for  any  given  element  varies  sensibly  as  the 
square  of  the  speed,  and  hence,  for  given  revolutions,  as  the 
square  of  the  radius.  Hence  the  entire  thrust  will  vary 
sensibly  as  the  integration  of  each  element  of  area  multiplied 
by  the  square  of  its  radius,  or  as  the  moment  of  inertia  of  the 
blade  area  about  the  axis.  Denote  this  moment  for  an 
elliptical  blade  by  70,  and  for  another  blade  of  equal  area  but 
different  form  by  /.  Then  the  form  of  blade  being  the  only 
variable  element  between  the  two  propellers,  we  should 
expect  that  the  values  of  the  thrust  would  be  sensibly  in  the 
ratio  of  /„  to  /,  or,  since  the  areas  are  equal,  in  the  ratio  of  the 
squares  of  the  radii  of  gyration.  It  should  be  remembered 
that  this  ratio  is  intended  simply  to  provide  for  the  results  of 
a  change  inform,  the  coefficient  m  being  intended  to  provide 
for  the  results  of  a  change  in  area.  A  change  of  form,  how- 
ever, implies  a  change  of  area  in  certain  parts  of  the  blade, 
and  hence  a  probable  change  in  the  value  of  the  coefficient 
a,  and  hence  a  slight  departure  in  the  value  of  the  thrust 
from  proportionality  to  the  moment  of  inertia  or  square  of 
the  radius  of  gyration. 


256  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

A  special  examination  of  this  point  seems  to  indicate, 
however,  that  the  error  should  not  be  sensible  so  long  as  the 
blade  is  in  general  oblong  or  oval  in  form,  while  if  distinctly 
widest  near  the  tips  the  actual  gain  in  thrust  will  be  some- 
what less  than  that  indicated  by  the  ratio  of  the  moments  of 
inertia.  We  should  be  hardly  justified,  moreover,  in  extend- 
ing our  efficiency  data  to  blades  of  this  form ;  so  that  it  may 
be  understood  as  a  general  limitation  that  the  blades  to 
which  these  equations  and  methods  are  to  be  applied  should 
be  of  a  generally  oblong  or  oval  form,  and  that  if  they  are 
markedly  trapezoidal,  with  the  broad  end  at  the  tip,  the  use 
of  these  methods  will  be  attended  with  more  uncertainty. 
At  the  same  time  if  an  estimate  relative  to  such  blades  has 
to  be  made,  and  no  means  of  more  direct  comparison  are 
available,  the  use  of  the  relations  indicated  will  probably 
furnish  the  best  estimate  obtainable  under  the  circumstances, 
the  true  values  of  the  forces  and  work  involved  being  prob- 
ably somewhat  smaller  than  those  given  by  the  estimate.  In 
a  large  number  of  applications  of  these  formulae,  made  under 
the  author's  direction,  for  the  purpose  of  analyzing  the  per- 
formance of  propellers,  they  were  applied  to  all  varieties  of 
form,  including  many  of  extreme  proportion.  The  general 
closeness  of  correspondence  and  the  consistency  of  the  results 
were  much  greater  than  had  been  anticipated,  and,  so  far  as 
this  investigation  indicated,  these  formulae  and  methods 
would  seem  likely  to  furnish  a  satisfactory  degree  of  accuracy 
for  most  designing  purposes. 

In  order  to  introduce  this  influence  due  to  variation  in 
form  into  our  equations,  let 

7    _  p**A_       ^_       . 

70  ==  p;A  ==  P:  ~  *' 


PROPELLED   DESIGN. 


257 


>eing  the  radius  of  gyration  of  the  blade  area  about  the 
axis.  The  value  of  i  may  be  determined  by  an  approximate 
integration  as  follows.  The  value  of  /  is  evidently  repre- 
sented by  the  integral 


in  which  b  is  the  breadth  at  radius  r.  The  approximate 
value  of  this  for  any  given  blade  may  be  obtained  in  the 
manner  indicated  in  the  following  form,  referring  to  the  pro- 
peller of  Fig.  75 : 


r 

r* 

b 

br* 

.2 

.04 

19 

.76 

•3 

.09 

22 

1.98 

•4 

.16 

23-5 

3-76 

•5 

•25 

24 

6.00 

.6 

.36 

24 

8.64 

•7 

•49 

23-3 

11.42 

.8 

.64 

21.2 

13-57 

•9 

.81 

17.0 

13-77 

I.O 

I  00 

5-o 

5-00 

179.0 

64.90 

12.0 

2.88 

I67 

62.02 

•3714 

The  column  headed  r  gives  the  fractions  of  the  entire 
radius  at  which  the  breadths  b  are  measured.  The  next 
column  gives  the  squares  of  these  values.  The  third  column 
gives  the  breadths  of  the  blade  at  the  corresponding  radial 
distances.  The  fourth  column  gives  the  products  br*,  which 
are  integrated  here  by  the  trapezoidal  rule  as  sufficiently 
exact  for  the  purpose  in  view.  The  column  is  summed,  and 
from  the  result  the  half  sum  of  the  end  values  is  subtracted, 
giving  the  value  62.02.  This  quantity  is  proportional  to  /, 


258  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

the  omitted  factor  being,  as  is  readily  seen,  rs  -~  10,  whence 

7  =  6.2O2r3. 

The  area  of  the  blade  is  determined  from  column  3  by  a 
similar  integration,  the  omitted  factor  here  being  r  -7-  10. 
Hence 

A  =  i6.?r. 

It  is  found  in  practice  more  convenient  to  use  the  ratio 
p  -7-  p02  rather  than  7-r-  70.      In  this  case,  therefore,  we  have 

/        6.202r8 


and  /=  .37147-^4. 

By  a  similar  proceeding  we  should  find  for  any  elliptical 
blade  with  the  same  relative  size  of  hub 

Po2  =  .357?",    .......     (3) 

and  7.  =  .357^  .......     (4) 

p2        .371  4^2 
Hence          ,•=_-==__  =  1.04. 

The  values  given  in  (3)  and  (4)  are  the  same  for  any  ellip- 
tical blade,  no  matter  what  the  surface-ratio,  provided  the 
diameter  of  hub  is  .2  that  of  the  propeller.  Denote  the  ratio 
of  hub  to  propeller  by  x.  Then  for  other  values  of  x  the 
following  empirical  formula  will  give  very  closely  the  values 
of  i  for  all  elliptical  blades  in  terms  of  that  for  x  =  .2  as 
unity: 

*•=  i  +  i.i(*-  .20)  ......     (5) 

For  blades  of  other  form  p2  may  be  found  by  an  integra- 
tion similar  to  that  shown  above,  and  i  thus  determined. 
In  Figs.  70  to  79  are  shown  a  series  of  blades  of  the  same 


PROPELLER   DESIGN.  2$$ 

area,  and  for  x  =  .2,  but  of  varying  form,  with  their  values 
of  the  ratio  i.  These  illustrate  the  relation  of  i  to  form  for 
constant  value  of  x>  and  will  aid  in  making  an  estimate  of 
its  value  independent  of  the  detailed  computation. 

Introducing,  therefore,  the  factor  i  into  (2),  we  have 

U  =  (p'N'W  \iklw) (6) 

Influence  due  to  Thickness  of  Blades. — On  this  subject  but 
little  information  is  available.  We  know  in  general  that  an 
increase  of  thickness  will  increase  /,  and  hence  decrease 
thrust  and  efficiency.  Data,  however,  are  lacking  for  a 
quantitative  estimate  of  the  amount  of  such  influence.  So 
long  as  the  thickness  is  not  unusual  the  results  should  not 
show  any  marked  variation  due  to  this  cause,  and  we  shall 
not  further  refer  to  this  influence.  We  should  remember, 
however,  that  a  saving  in  thickness  so  long  as  rigidity  and  the 
necessary  strength  are  not  sacrificed  is  always  desirable  on 
the  score  of  efficiency. 

GENERAL   REMARKS. 

We  therefore  consider  (6),  together  with  the  equations, 
diagrams,  and  tables  relating  to  m,  k,  and  /,  and  the  efficiency 
information  as  derived  in  §  36  and  shown  in  Figs.  58  or  71, 
as  our  fundamental  data  for  propeller  design. 

In  discussing  the  application  of  these  data  to  actual  cases 
involving  the  various  changes  designated  under  the  headings 
(i),  (2),  (3),  and  especially  those  whose  effects  are  represented 
by  the  coefficients  i  and  ;;/,  the  assumptions  used  have  been 
necessarily  to  some  extent  hypothetical,  and  the  experimen- 
tal basis  for  a  determination  of  the  values  of  these  coefficients 
is  unfortunately  very  meager.  If,  however,  the  admissible 


260  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

changes  be  limited  to  (i),  that  is,  to  changes  in  dimension 
only,  with  constant  shape  of  blade  and  area-ratio,  then  m  and 
/are  both  I,  and  we  simply  reproduce  Froude's  experimental 
results  as  applied  to  such  a  series  of  propellers.  The  general 
agreement  between  the  results  given  by  intelligent  use  of  the 
method  thus  restricted,  with  successful  experience,  indicates 
that  it  may  be  employed  with  a  high  degree  of  confidence  in 
its  reliability,  and  that  in  any  event  it  will  be  much  better 
than  uncertain  comparison  between  the  results  of  experiment 
where  the  accuracy  of  the  data  may  be  called  in  question,  or 
than  any  method  which  involves  a  less  detailed  analysis  and 
representation  of  the  different  variables  involved  in  the 
problem.  The  introduction  of  the  coefficients  m  and  i  into 
the  method  is  for  the  purpose  of  providing  a  greater  elasticity 
in  the  choice  of  form  and  area-ratio,  without  a  serious  loss  of 
general  reliability.  While  undoubtedly  the  method  may  be 
used  with  more  confidence  when  form  and  area-ratio  are  con- 
stant and  the  same  as  in  the  model  propellers,  yet  for 
changes  of  moderate  amount  in  these  features  the  effects  must 
be  very  closely  represented  by  the  coefficients  i  and  ;;/  as 
given,  and  no  serious  departure  from  reliable  results  will  be 
thereby  introduced.  In  any  event  where  reliable  data  from 
more  direct  comparison  in  such  cases  is  not  available,  this 
method  will  probably  furnish  the  most  reliable  results  to  be 
found  under  the  circumstances. 

Again,  as  will  appear  more  fully  by  illustrative  examples, 
this  general  method  does  not  take  the  place  of  judgment  and 
experience.  So  far  as  the  method  itself  is  concerned,  there 
may  result  an  indefinite  number  of  propellers,  each  more  or 
less  completely  fulfilling  the  conditions  imposed,  and  in  the 
discrimination  between  these  full  scope  may  be  found  for  the 


PROPELLER   DESIGN. 


26l 


use  of  judgment  and  experience.  It  is  intended  rather  as  an 
aid  to  the  intelligent  use  of  experience,  and  to  the  systematic 
analysis  of  the  results  of  trial  data. 

50.  PROBLEMS  OF  PROPELLER  DESIGN. 

Formula  (6),  §  49,  relates  simply  to  the  useful  work  of  a 
propeller  operating  in  undisturbed  water.  In  the  actual  case 
the  propeller  operates  in  the  forward  wake.  In  Chapter  III 
we  have  discussed  this  wake,  and  shown  that  we  may  substi- 
tute for  the  actual  turbulent  wake  a  uniform  stream  of 
velocity  v9  giving  a  true  slip  S3  =  5X  +  ^0  >  anc*  a  sPeed  of 
advance  of  the  propeller  relative  to  the  water  in  which  it 
works,  of  PN  —  5a  —  PN  —  Sl  —  v9  =  u  —  v9.  Now  in  order 
to  connect  the  propeller  in  undisturbed  water  with  the  actual 
case,  we  assume  that  for  a  given  propeller  at  given  revolu- 
tions working  in  the  wake  of  velocity  z>0  and  developing  a 
thrust  of  T  and  a  speed  relative  to  the  wake  of  (u  —  z/0),  the 
useful  work  and  efficiency  will  be  the  same  as  if  it  were  work- 
ing in  undisturbed  water  at  the  same  revolutions  and  true 
slip,  and  hence  with  an  equal  speed  of  advance  (u  —  z/0). 
That  is,  referring  to  Fig.  70,  we  must  consider  the  useful 
work  of  our  formula  as  based  on  the  true  propeller  efficiency 
and  not  the  apparent,  or  on  the  speed  of  advance  of  the  pro- 
peller through  the  water  in  which  it  works,  and  not  relative 
to  still  water.  Hence  the  useful  work  thus  defined  will  be 
represented  by  C^E  =  T(u  —  7>0),  and  not  by  CE  =  Tu. 

It  is  therefore  the  work  C^E  which  must  be  substituted 
in  the  formula,  §  49,  (6). 

The  following  will  therefore  be  the  most  natural  order  of 
procedure: 

The  I.H.P.   being  given  or  AE,   we  assume  AB.     This 


262  RESISTANCE   AND   PROPULSION   OF  SHIPS. 

gives  us  BE.  We  then  select  or  assume  the  true  efficiency 
at  which  we  propose  to  work,  or  else  fix  the  true  slip  and 
pitch-ratio  which  together  determine  the  efficiency.  Then 


A  6  C     C,  F       E 

FIG.  70. 

BE  multiplied  by  this  efficiency  gives  C^,  the  useful  work 
to  be  substituted  in  our  formula.  Again,  it  may  arise  that 
instead  of  the  I.H.P.  we  have  the  E.H.P.,  or  assume  the 
propulsive  coefficient,  and  thus  pass  from  AE  to  CF  direct. 
Then  CF  multiplied  by  hull  efficiency  =  CtE,  the  useful 
work,  as  before.  As  already  noted,  unless  special  information 
is  available,  the  hull  efficiency  is  usually  taken  as  unity. 

We  must  next  determine  the  value  of  pN. 

Let  u  =  speed  in  knots.     Then 

101.32*  =  speed  in  feet  per  minute. 


. 

But  from  §  44,  (4), 

(l  -  s,)  =  (l  +  w)(l  -  s,). 


Hence  pN  — 


This  determination  calls  for  an  estimate  of  the  wake 
factor  w.  Unfortunately  the  information  available  as  a  basis 
for  such  estimate  is  very  meager.  Froude  in  connection  with 
the  paper  on  propellers  previously  referred  to  gives  a  series 
of  values  for  ships  of  various  characteristics,  and  beyond  this 
next  to  no  information  is  available  in  the  general  literature  of 
the  subject.  Adding  to  these  a  few  other  known  values, 


PKOPELI.ER   DESIGN. 


263 


Prof.  McDermott*  has  connected  them  with  the  character- 
istics of  the  ships  by  empirical  formulae,  as  follows: 
Let  w  denote  wake  factor; 

p      "         prismatic  or  cylindrical  coefficient; 
;;/     "         midship-section  coefficient; 
L     "        length  in  feet. 
Then 

w  =  .  i6( — Z*  —     .6J  for  single-screw  ships; 

w  =  .13! — i^  —  I*lJ  for  twin-screw  ships. 

These  equations  represent  simply  the  available  data,  and 
as  this  is  relatively  meager,  care  must  be  exercised  in  their 
use.  Where  no  special  information  is  otherwise  obtainable, 
however,  they  will  probably  serve  to  give  a  fair  approxima- 
tion to  the  value  desired.  Further  suggestions  on  this  point 
as  well  as  on  several  others  relating  to  the  choice  of  values 
for  the  various  quantities  involved  will  be  given  in  the  fol- 
lowing section.  We  will  now,  without  more  delay,  illustrate 
the  application  of  our  formulae,  diagrams,  and  tables  to 
actual  problems. 

We  first  repeat  in  collected  form,  for  convenience,  the 
principal  equations  which  we  may  have  occasion  to  use : 


100' 


10 


(i) 


(3) 


*  Transactions    Society   of    Naval    Architects   and    Marine    Engineers, 
vol.  iv.  p.  164. 


264 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


(I  +  «0  = 


i  —  V 

d   4.  w\(i    —    c); 

w 

(5) 

(6) 

I    -    V 

(.034  4-  .8cx.y  —  .68$')  ;  . 

(7) 

(  v,  —  .17"):   . 

(to 

w 


—  .  i6(—  L^  —    .6]  for  single  screw-ships;       (9) 

/  v,  \ 

ze/  =  .13^ — Lfr —  1. If  for  twin-screw  ships ;  .     (10) 

m  is  estimated  by  aid  of  Fig.  69,  p.  253; 
i  is  determined  by  computation  or  by  judgment, 

guided  by  Figs.  72  to  81,  and  §  49  (5). 
The  values  of  k,  /,  and  of  101.32^  may  also  be  taken  from 
Tables  I,    II,    III,    for  the   values   of   slip,    pitch-ratio,    and 
speed  given,  and  by  interpolation  for  intermediate  values. 

TABLE  I.   VALUES  FOR  FACTOR  k. 


Slip. 

k 

Slip. 

k 

.10 

.108 

•25 

•193 

.  II 

.114 

.26 

.197 

.12 

.121 

•27 

.202 

•13 

.127 

.28 

.206 

.14 

•133 

.29 

.210 

•15 

.140 

•30 

.214 

.16 

•145 

•31 

.218 

•17 

•151 

.32 

.222 

.18 

•157 

•33 

.226 

.19 

.162 

•34 

.229 

.20 

.168 

•35 

•233 

.21 

•173 

.36 

.236 

.22 

.178 

•37 

•239 

•23 

•183 

•38 

.242 

.24 

.188 

•39 

•245 

PROPELLER   DESIGN. 


TABLE   II.      VALUES   FOR   FACTOR   /. 


265 


Pitch-ratio. 

/ 

Pitch-ratio. 

/ 

Pitch-ratio. 

/ 

.90 

.941 

.40 

•544 

-90 

•356 

.92 

.917 

•42 

•534 

.92 

•351 

•94 

.894 

•44 

•524 

•94 

•345 

.96 

.872 

.46 

•515 

.96 

•340 

.98 

•  850 

.48 

.506 

.98 

•335 

1.  00 

•  830 

•50 

•497 

2.00 

•330 

.02 

.810 

•52 

.488 

2.02 

•325 

.04 

.792 

•54 

•479 

2.04 

-320 

.06 

•773 

•56 

.471 

2.06 

•315 

.08 

•756 

•58 

•463 

2.08 

•  310 

.10 

•739 

.60 

•455 

2.10 

•  306 

.12 

•723 

.62 

•447 

2.12 

•  302 

.14 

.707 

.64 

.440 

2.14 

•297 

.16 

.692 

.66 

•432 

2.16 

•293 

.18 

.677 

.08 

•425 

2.18 

.289 

.20 

.663 

•70 

.418 

2.20 

.285 

.22 

.650 

•72 

.411 

2.22. 

.280 

.24 

.636 

•74 

•405 

2.24 

.276 

.26 

.624 

.76 

•398 

2.26 

.272 

.28 

.611 

.78 

•392 

2.28 

.269 

•30 

•599 

.80 

.386 

2.30 

.265 

•32 

.588 

.82 

•379 

2.32 

.261 

•34 

•576 

.84 

•373 

2-34 

•257 

.36 

.565 

.86 

•  368 

2.36 

•254 

1.38 

•555 

1.88 

•  362 

2.38 

-250 

TABLE  III.   GIVING  FEET  PER  MIN.  CORRESPONDING  TO  KNOTS, 
OR  VALUES   OF    IOI.3?/. 


u 

101.  3«. 

u 

ioi.3«. 

10 

1013 

25 

2533 

II 

IH5 

26 

2635 

12 

1216 

27 

2736 

13 

1317 

28 

2837 

14 

1419 

29 

2939 

15 

1520 

30 

3040 

16 

1621 

31 

3MI 

17 

1723 

32 

3243 

18 

1824 

33 

3344 

19 

1925 

34 

3445 

20 

2027 

35 

3547 

21 

2128 

36 

3648 

22 

2229 

37 

3749 

23 

2331 

38 

3851 

24 

2432 

39 

3952 

266 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


1.0      1.1      1.2      1.3      1.4      1.5      1.6       1.7      1.8      1.9      2.0      2.1      2.2      2.3      2.4      2.5 

PITCH  RATIO 
FIG.    71. 


PROfELLER  DESIGN. 


267 


V.. 


-_4- 


J 


FIG.  72. 


FIG.  73. 


\ 


i=1.04 


J 


FIG. 


FIG.  75- 


268  RESISTANCE  AND    PROPULSION   OF  SHIPS. 


1=1.16 


FIG.  76. 


FIG.  77- 


V 


FIG.  78. 


PROPELLER  DESIGN. 


269 


j  i- 


FIG.  80. 


FIG.  81. 


EXAMPLES. 

(a)  As  the  first  example  let  us  take 
I.H.P.  =4600; 

speed  =  u  =  16  knots; 

friction  power  =  13  per  cent  =  598  H.P.  (AB,  Fig.  70). 
Hence 

propeller  power  =  4002  H.P.  (BE,  Fig.  70). 

Next  from  the  data  given  suppose  w  found  or  estimated 
as  .14.  We  will  also  select  .26  as  the  proposed  value  of  the 
true  slip  or  j,.  We  then  find  from  (6) 

i  —  s,  =  .844     and     j,  =  .156. 


Hence 
and 


16  X  101.3 
PN= ^T     -  =  1922; 


.844 
p'N'  =  19.22. 

Now  supposing  neither/  nor  TV  fixed,  we  may  proceed  as 
follows:  We  first  select  a  pitch-ratio  of,  let  us  say,  c  =  1.26. 


270  RESISTANCE  AND    PROPULSION   OF  SHIPS, 

We  will  also  assume  in  this  problem  that  the  blades  are 
elliptical,  with  area-ratio  .36,  so  that  both  i  and  m  are  I. 
From  Tables  I  and  II  and  Fig.  71  we  then  find 

£=.197; 

/=.624; 

e,  =  .68. 
Hence 

useful  work  U=  .68  X  4002  =  2721  (C,E,  Fig.  70). 
Then  substituting  we  have 

2721  =  (i9.22)W/2  X  .197  X  .624; 
whence  d*=    3.141; 

and  d  —  17.72  ; 

N=S6.i. 

(fr)  Let  the  conditions  remain  the  same  as  above,  except 
as  regards  the  blades.  Let  the  area-ratio  be  .48  and  the 
shape  such  that  i  is  taken  =  1.03.  From  Fig.  69  it  appears 
that  we  may  take  m  about  I.I.  Hence  we  have 

272 1  =  (i9.22)V2  X  1.03  X  .  197  X  .624  X  1. 1 ; 
whence  d"1  =    2.766; 

and  d=  16.6; 

N=92. 

(c)  Suppose  with  the  same  data  as  in  (a)  we  fix  both 
pitch  and  diameter  and  propose  to  find  the  necessary  shape 
of  blade  or  amount  of  area.  Thus  let  /  =  24  feet  and 
d=  1 8  feet.  Then  with  the  same  value  of  pN  we  find 
N =  80. i.  Also  c  =  1.33,  and  we  have 


PROPELLER   DESIGN. 

k  =  .  197,  as  before; 


271 


and          £7=2725. 

Substituting  and  solving  for  the  combined  factor  (tin),  we 
then  find 

2725 


(im)  - 


=  1-03, 


7100  X  3-24  X  .197  X  .582 

This  may  be  realized  with  an  elliptical  blade  of  area-ratio 
about  .40  or  with  an  area-ratio  of  .36  and  shape  similar  to 
Fig.  73  or  by  any  desired  combination  of  the  two  features 
area  and  shape  such  that  the  product  im  shall  equal  1.03. 

(d)  Given  the  same  data  as  in  (a),  let  us  fix  N  at   no. 
Then  with  the  same  value  of  pN  we  have 

/==  1922  +  110=  17.5. 
Let  us  now  assume  for  trial  a  diameter  of  16  feet.     Then 

c=  17-5  +  16=  i.i; 
and  we  have 

k  =  .  197,  as  before; 

/=  -739; 

'.  =  -677; 

and       U  =  2710. 

Substituting  for  (im)  as  in  (c),  we  have 

2710 


(im)  = 


=  1.024. 


7100  X  2.56  X  .197  X  .739 

A  slight  increase  of  area-ratio  or  a  slight  filling  out  of  the 
form  at  the  outer  end  of  the  blade  will  therefore  fulfil  the 
indications  here  given. 


2/2  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

The  propellers  thus  far  considered  have  been  implicitly 
supposed  to  have  four  blades.  To  illustrate  the  application 
of  the  equations  to  propellers  of  two  or  three  blades  we  may 
take  the  following: 

0)  Let  I.H.P.  =  5000; 

u  =  20; 

Wp=&7  X  5000  =  4350; 
w  —  .08; 


Then  (i  —  s,)  =  .81, 

2027 

and  pN  =  -—- 

whence       (/W)  =  25.03. 

Take,  also,         pitch-ratio  =  1.4; 
number  of  blades  =  3  ; 
area-ratio  =  .30; 
i=  i.  oo. 

Then  we  find 


/=  .5441 
m  =  .92; 
*,  =  .68; 
17=  2958. 

Then,  substituting,  we  have 

d     ~~  15681  x  .193  x  .544  x  .92  = 


PROPELLER   DESIGN.  273 


Whence  we  find 


d  =  14; 
/  =  19-6; 
N=  128. 

As  an  example  of  a  two-bladed  propeller  we  may  take  the 
following: 

(/)  I.H.P.  =  8; 

*==7; 

JF,  =  .85  X  8  =  6.8; 
w  =  .  10; 
s,  =  .30. 

Then  (I  -  -0  =  .77, 

and  pN  =921; 

whence       (p'N')=  9.21. 

Take,  also,  pitch-ratio  =  i.i; 

number  of  blades  —  2, 

area-ratio  =  .20; 
i  =  i  .  1  6. 


Then  we  find 


k=  .214; 

/=  -739; 

=  .72; 

et=  .66; 


Then,  substituting,  we  have 


d'*  =  - 


4.49 


781  X  1. 16  X  .214  X  -739  X  .72 

Whence  we  find 

d—  2.09; 

/>  =  2.3; 
A'  —  400. 


=  -0435- 


274  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

(g)  As  a  further  illustration  of  special  design  let  us  take 
the  case  of  a  tugboat.  If  the  design  is  based  on  the  condi- 
tions of  use  we  must  naturally  take  the  speed  low  and  the 
slip  high. 

Let  I. H. P.  -300; 

u  =  7\ 

Wp=  .83  X  300  =  249; 
w  =  .12; 
st  -  .42. 

Then  (i  —  s,)  =  .65; 

(pN)=  1091; 
and          (p'N1)  =  10.91. 

Take  also  pitch-ratio  =  1.2  ; 

number  of  blades  —  4 ; 

area-ratio  =  .54; 
i  —  i.  20. 

Then  we  find 

£  =  .2  54  (from  (7)); 

/=.66s; 
m  =  1.15; 
e^  =  .60  (by  extension  of  Fig.  71); 

U—  149-4- 
Then  substituting  we  have 

149.4 
1299  X  1.15  X  1.2  X  .254.x  -663 

Whence  we  find 

d=  7-04; 

/  =  8.45; 
N=  129. 

The  usual  operations  involved  may  be  somewhat  general- 
ized as  follows: 


PROPELLER   DESIGN. 


275 


Given  I.H.P.,  speed  u,  friction,  wake  factor  w,  and  true 
slip  st. 

Whence  IV p  or  BE,  Fig.  70,  slt  pN,  and  k. 
Then  (i)  Given  neither/  nor  N. 

Assume  pitch-ratio  c,  i,  and  m. 

Thence  find  et,   U,  and  /. 

Thence  by  (i)  find  d. 

Thence/  and  N. 

(2)  Given  either/  or  N. 
Thence  the  other  N  or  p. 
Assume  d. 

Thence  find  c,  e^  U,  and  /. 

Thence  by  (i)  find  (im)  and  divide  between  the 

two  according  to  choice. 
Thence  follow  shape  and  area-ratio. 

(3)  Given  d. 
Assume  c  or/. 

Thence  find/  or  c  and  N,  also  e^  Uy  and  /. 
Thence  as  in  (2). 

By  an  inversion  of  the  above  processes  these  equations 
may  be  used  for  the  analysis  of  a  given  set  of  trial  data,  and 
thus  for  the  determination  of  the  values  of  .$•„  and  w  in  the 
given  case.  The  use  of  the  equations  for  this  purpose 
assumes  them  as  applicable  to  the  given  propeller,  and  hence 
we  should  expect  the  results  to  be  generally  more  reliable  the 
more  closely  the  actual  propeller  approaches  to  the  standard 
form  and  area.  The  operation  is  as  follows: 

(//)  Given  I.H.P.,  /,  N,  d,  it  in,  /,  and  j,,  and  assume 
engine-friction. 

Thence  find   Wp  and  pitch-ratio. 

k 

Thence  from  (i)  the  value  of  — . 


276 


RESISTANCE   AND    rROPULSlON   OF  SHIPS. 


In   Fig.   82   we  have  plotted  the  values    of    '  -  on  pitch- 

^i 

ratio    as    abscissa,    each    curve    for    constant    value   of   s^   as 
shown.     We  have  therefore  simply  to  take  on  this  diagram 


.110 
.100 


1.0      1.1      1.3      1.3      l.i      1.5 


1.6      1.7      1.8 

PITCH   RATIO 

FIG.    82 


2.1      2.2      2.3      2.4 


the  point  corresponding  to  the  given  value  of  pitch-ratio  and 
k  -L-  ev  The  location  of  this  point  relative  to  the  curves  for 
constant  slip  will  then  show  the  corresponding  value  of  s9. 

Thus,  for  example,  taking  the  data  of  (a)  except  st  and  wt 
we  find 


k 


4002 


7100  X  3.141  X  .624 


=  .288. 


PROPELLER    DESIGN. 


2/7 


Entering  the  diagram  on  pitch-ratio  1.26  we  find  .26  as. 
the  nearest  value  of  the  slip,  corresponding  to  the  value 
originally  taken  in  (a).  By  substitution  in  (6)  we  then  find 
w  =  .14,  while  from  Fig.  7 1  ra  =  .68. 

As  a  further  example  we  may  take  the  following  data 
from  the  U.  S.  cruiser  Detroit: 

We  have  given  as  mean  results  for  the  two  propellers: 

D  =       11; 
p  =       13; 

pitch-ratio  =         1 .  1 8  ; 
I. H. P.  -2577; 
N  =     1 70. 1 ; 
18.71; 
.142; 

.305; 
.90; 
1.09; 

.677. 

Engine-friction  is  taken  at  14  per  cent. 
Hence  Wp  =  2216. 
Then  we  have 
k  _  2216 

7,  ~  i. 21  x  10808  x  1.09  x  .677  x  .9  =  *255" 

From  Fig.  82  we  thence  find  st  =  .215. 

.858 
Hence  i  -f-  w  =        -  =  1.093, 

and  w  —  .093  ; 

while  from  Fig.  71  e^  =  .688. 

The  use  of  the  equations  in  this  manner  in  a  large  number 
of  cases  carried  out  under  the  direction  of  the  author 
showed  that  it  is  somewhat  difficult  to  satisfactorily  determine 


u  = 


area-ratio  = 


m  = 


278  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

the  value  of  w  from  such  analysis  alone.  This  is  not  so 
much  due  to  inaccuracy  in  the  equations  as  to  the  fact  that  a 
slight  change  or  error  in  the  data  will  make  a  relatively  much 
larger  error  in  the  value  of  iv.  This  arises  from  the  fact  that 
the  value  of  w  is  determined  from  the  quantity  (i  -|-  w). 
Now  an  error  of  moderate  amount  in  the  data  or  in  the  suit- 
ability of  the  equations  to  the  data  might  give,  for  example, 
an  error  of  2  or  3  per  cent  in  the  value  of  (i  -f-  zv),  making, 
for  example,  its  value  1.12  when  it  should  be  i.io.  This 
would  make  a  difference  of  .02  in  w,  or  an  error  of  20  per 
cent  in  excess  of  the  true  value.  So  far  as  ordinary  applica- 
tions are  concerned,  however,  and  especially  so  far  as  the 
problem  of  design  is  involved,  it  is  the  factor  (\-\-w)  and 
not  w  which  is  used,  and  hence  changes  of  relatively  large 
percentage  amount  in  w  will  have  comparatively  small  influ- 
ence on  the  solution  of  such  problems.  It  may  be  also  men- 
tioned that  a  comparison  of  the  values  of  w  determined  in 
this  manner  with  those  given  by  equations  (9)  and  (10)  and 
by  general  estimate  showed  in  nearly  all  cases  that  a  differ- 
ence of  a  few  per  cent  in  the  friction  of  the  engine,  or  more 
especially  of  a  very  small  amount  in  the  pitch  of  the  pro- 
peller, would  be  sufficient  to  account  for  the  difference  in  the 
values  of  w.  The  uncertainty  as  to  the  real  value  of  the 
pitch  in  cases  of  variable  pitch  has  been  considered  in  §  42  ; 
and  bearing  these  facts  in  mind  as  well  as  the  possibilities  of 
slight  errors  of  measurement  in  all  cases,  and  the  further 
uncertainty  regarding  engine-friction,  it  would  appear  that 
the  degree  of  fulfilment  is  closer  than  might  naturally  have 
been  expected,  and  that  for  problems  of  design  the  equations 
and  methods  may  be  used  with  a  high  degree  of  confidence 
in  the  results. 


PROPELLER   DESIGN. 


279 


The  influence  of  an  incorrect  estimate  of  w  in  a  problem  of 
design  may  be  illustrated  by  the  following  example: 

(/')  In  problem  (a)  above,  suppose  .  14  to  be  the  correct 
value  of  «',  but  let  its  estimated  value  be  taken  as  .09.  Then 
with  the  same  assumed  value  of  s^  we  shall  have 

i  —  s,  =  .807; 
pN  —  2009. 

Following  the  work  through  as  in  (a),  we  should  find  the 
following  results: 

d  =  16.6, 
P  =  20.9, 


instead  of  those  found  in  (a). 

Now  suppose  this  design  accepted  and  the  propeller  fitted 
to  the  ship.  We  may  then  ask  two  questions:  (i)  What  will 
be  the  change  in  efficiency  and  what  the  power  necessary  at 
the  desired  speed  of  16  knots?  or,  (2)  What  will  be  the  re- 
sultant speed  if  the  power  4002  is  delivered  to  the  propeller 
as  in  (a)? 

We  first  assume  the  speed  of  16  knots  attained  with  a 
value  of  C,E  =  2721,  as  in  (a).  We  then  have,  by  substitution 
in  (i), 


2721  = 


6.  624. 


In  this  equation  (p'N')  and  k  are  both  unknown,  but  both 
dependent  on  st.      Hence  solving,  we  have 

(p'N'}'k=  1582. 


280  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

We  must  now  find  by  a  process  of  trial  a  value  of  J3  sue 
that  the  resulting  values  of  k  and  p'N'  (using  of  course  the 
true  value  .  14  for  w)  will  give  for  (p'N')*fc  the  value  1582,  or 
a  sufficiently  close  approximation  to  such  value.  For  the 
value  s,  —  .26  assumed  in  (a)  we  have  k  =  .197  and  (p'N')* 
—  7100.  Whence  (p'N'^k  —  1399.  The  value  1582  re- 
quired, indicates  a  value  of  s^  larger  than  .26,  and  one  or  two 
trials  are  sufficient  to  show  that  the  desired  result  is  very  near 
to  ,28,  for  which  (pfN')*k  —  1587.  We  may  take,  therefore,. 
as  the  result 

s%  =  .28; 

pN  =  1975; 

N=  94.5; 

e,  =  .673,  from  Fig.  71  , 
-7-67.3  =4043- 


According  to  these  results,  therefore,  the  revolutions 
would  be  slightly  less  than  the  96. 1  expected,  the  efficiency 
would  be  decreased  by  .007,  and  there  would  be  required  41 
additional  H.P. 

If  we  next  suppose  but  4002  H.P.  delivered  to  this  pro- 
peller and  inquire  as  to  the  resulting  speed,  we  have  a 
problem  requiring  for  its  solution  a  resistance-curve  for  the 
ship.  For  all  practical  purposes,  however,  we  may  assume 
the  same  loss  of  efficiency  .007,  and  that  power  varies  as  the 
cube  of  the  speed.  This  would  give  for  the  ratio  of  the  power 
in  the  two  cases  the  value 

.68  -f-  .673  =  i. 01, 
and  therefore 

Ratio  of  speeds  =  v'l.oi  =  1.003. 


PR O PE LLEK    D ESIGN. 


28l 


Hence  with  4002  H.P.  at  the  propeller  in  the  case  assumed 
we  should  have  a  speed  of 

u  =  16  -^  1.003  =  15-95. 

or  a  resulting  loss  of  .05  knot. 

In  a  similar  fashion  if  we  suppose  w  overestimated  by  .05, 
or  taken  at  .  19  instead  of  .14,  we  may  carry  the  work  through > 
finding  for  the  propeller  the  following: 

pN=  1840; 
d  =  18.9; 
^  =  23.8; 

^=77.3. 

Then,  as  before,  we  should  find  that  such  a  propeller  ap- 
plied to  the  ship  in  (a)  with  w  =  .  14  instead  of  .  19  would  give 
rise  to  the  following  values : 

*.  =  -24; 
pN=  1872; 

^=78.7; 

e^  =  68.5,  from  Fig.  71  ; 
Wp  =  2721  -7-685  =  3972. 

In  the  same  manner  as  above  we  should  also  find  that  with 
4002  H.P.  delivered  to  this  propeller  the  speed  would  be  in- 
creased by  about  .04  knot  or  to  16.04  knots. 

These  illustrations  of  the  influence  of  an  error  in  the  esti- 
mation of  w  may  perhaps  be  taken  as  extreme  cases,  for  we 
should  under  all  usual  conditions  be  able  to  estimate  w  within 
.05 ,  and  it  thus  appears  that  the  errors  resulting  from  an  incorrect 
estimate  of  w  are  in  themselues  not  likely  to  be  serious,  cspc- 


282  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

cially  if  the  propeller  is  designed  to  work  at  or  near  its  maxi- 
mum efficiency.  Some  further  discussion  of  these  points  will 
be  found  in  the  following  section. 

Additional  Methods  of  Propeller  Design. — The  formulae  and 
methods  of  design  whose  applications  are  shown  in  the  present 
section  are  such  as  naturally  result  from  the  general  method 
of  treatment  developed  in  this  work.  The  same  original  data 
(that  of  Froude's  experiments)  may  be  used  in  a  variety  of 
other  ways  as  illustrated  by  Froude,*  Barnaby,f  Caird, \  and 
McDermott.§  There  have  been  long  in  use  also  briefer  and 
less  accurate  methods  of  screw-propeller  design  than  those  in- 
volving the  detailed  considerations  discussed  in  the  present 
work.  A  brief  reference  to  such  methods  may  here  be  made. 

The  value  of  pN  is  found  from  the  given  speed  u  and  an 
assumed  value  of  the  apparent  slip  j,  (see  (4) ).  The  values  of 
/and  A^are  then  selected  by  judgment. 

The  value  of  d  may  then  be  found  from  the  formula 


-nit 


I.H.P. 

T/wr 


in  which  Kl  is  usually  taken  between  18000  and  25000. 

It  is  readily  seen  that  this  is  equivalent  to  taking  the  value 
of  the  product  z,  k,  /,  m,  (i)  as  constant,  or  assuming  its  value 
by  judgment  between  limits  corresponding  to  those  specified 
for  Kr 


*  Transactions  Institute  of  Naval  Architects,  vol.  xxvn.  p.  250. 

f  Transactions  institution  of  Civil  Engineers,  vol.  cir.  p.  74. 

;  Transactions  Institute  of  Engineers  and  Shipbuilders  in  Scotland, 
1895-96,  p.  21. 

^  Transactions  Society  of  Naval  Architects  and  Marine  Engineers,  vol. 
IV.  p.  159- 


PROPELLER   DESIGN. 

The   helicoidal   area   A    is   sometimes  determined   by  the 
formula 


in  which  A',  may  vary  from  8  to  15. 

The  helicoidal  area  may  also  be  determined  by  the  formula 

Indicated  thrust       I.H.P.  X  33OOO  / 


in  which  K%  may  vary  from  1000  to  1500. 

Having  thus  determined  A,  the  resulting  values  of  the 
disk  area  and  of  the  diameter  d  may  readily  be  found  by  the 
use  of  standard  proportions. 

Since  helicoidal  area  should  vary  nearly  as  d*,  it  is  evident 
that  the  three  equations  above  are  mutually  inconsistent  in 
form,  though  when  used  with  judgment  and  with  abundance 
of  data  for  comparison  they  may  be  made  to  yield  good  re- 
sults. The  third,  relating  helicoidal  area  to  indicated  thrust, 
is  probably  the  most  satisfactory  of  the  three. 

The  disk  area  may  also  be  related  to  the  wetted  surface  of 
the  ship  by  the  formula 


Disk  area  = 


Wetted  surface 


in  which  A"4  is  usually  found   between    70  and  90,  while  for 
specially  high-powered  craft  it  may  fall  to  from  50  to  70. 

The  disk  area  may  also  be  related  to  the  area  of  midship 
sc -ction  by  the  formula 

Area  of  midship  section 
Disk  area  =  -  — — —  — , 


284  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

in  which  A"5  is  usually  found  between  2  and  2.5,  while  for 
specially  high-powered  craft  it  may  fall  to  from  1.2  to  2. 

The  disk  area  may  also  be  related  to  the  displacement  D 
by  the  formula 

Disk  area  =  — , 
K* 

in  which  Kt  is  usually  found  between  .8  and  I.I,  while  for 
specially  high-powered  craft  it  may  fall  to  from  .6  to  .8. 

In  formulae  relating  the  disk  area  to  the  ship  the  result 
found  is  considered  as  the  area  for  all  the  propellers,  one, 
two,  or  three,  as  fitted.  The  value  may  then  be  divided  as 
required,  and  the  diameter  immediately  found. 

The  methods  and  formulae  briefly  referred  to  in  these 
equations  involve  necessarily  a  large  element  of  judgment,, 
with  but  slight  opportunity  for  its  orderly  exercise.  They 
have,  furthermore,  no  provision  for  estimating  in  detail  the 
probable  results  due  to  variations  in  the  different  charac- 
teristics of  the  propeller.  With  the  more  detailed  methods 
presented  and  illustrated  in  the  present  chapter,  the  range  of 
judgment  is  somewhat  narrowed,  its  application  is  simpler, 
and  full  provision  is  made  for  the  orderly  estimate  of  the 
probable  results  due  to  changes  in  the  more  important  char- 
acteristics. 

51.  GENERAL  SUGGESTIONS  RELATING  TO  THE  CHOICE 
OF  VALUES  IN  THE  EQUATIONS  FOR  PROPELLER 
DESIGN,  AND  TO  THE  QUESTION  OF  NUMBER  AND 
LOCATION  OF  PROPELLERS. 

In  the  method  of  the  preceding  section  the  I.H.P.  is 
taken  as  a  part  of  the  fundamental  data.  This  implicitly 


PKOPELLER   DESIGN. 


285 


involves  values  of  the  augmentation  of  resistance  due  to  the 
propeller  and  of  the  wake  factor.  While,  therefore,  the  aug- 
mentation of  resistance  is  not  explicitly  involved  in  our  equa- 
tions, it  is  really  represented  by  the  assumed  or  determined 
I.H.P.  The  wake  factor  w  is  not  only  involved  in  the 
I  II. P.,  but  we  are  also  called  on  to  assign  to  it  a  definite 
value. 

The  principal  source  of  information  relative  to  both  the 
v.  ake  factor  and  the  factor  of  augmentation  (or  its  reciprocal 
"thrust  deduction,"  as  termed  by  Froude)  is  in  the  experi- 
ments on  models.  These  points  are  discussed  by  Mr.  R.  E. 
Froude  in  the  paper*  to  which  previous  reference  has  been 
made,  and  to  which  the  reader  may  be  referred  for  details. 
The  conclusions  were  that  for  both  primary  uses  of  the  data, 
viz.,  the  determination  of  the  standing  of  a  propeller  on  the 
scale  of  efficiency  and  the  relation  of  its  thrust  to  revolutions, 
such  sources  of  error  as  may  properly  seem  to  exist  appar- 
ently tend  to  neutralize  each  other,  so  that  we  have  to  deal 
not  with  a  single  error,  nor  with  several  tending  in  one  direc- 
tion, but  with  a  balance  of  errors  in  which,  though  the  indi- 
vidual values  may  be  sensible,  it  is  quite  likely  that  their 
combination  will  not  form  an  error  of  serious  amount.  Com- 
paring the  numbers  of  revolutions  for  the  models  with  those 
for  the  full-sized  ships,  Mr.  Froude  states  that  the  model 
data  on  the  average  were  found  to  overestimate  the  revolu- 
tions of  twin-screws  by  about  2  or  3  per  cent,  and  to  under- 
estimate those  of  single-screws  by  about  the  same  amount. 

As  shown  by  example  (*'),  §  50,  the  result  of  an  under- 
otimate  of  w  when  the  design  is  carried  through  for  a  given 


*   1  r.msactions  Institute  of  Naval  Architects,  vol.  xxvu.  p.  250. 


286  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

pitch-ratio,  as  in  (a),  will  be  to  obtain  a  propeller  too  small 
and  with  a  larger  true  slip  than  that  assumed,  while  an  over- 
rated  w  will  give  rise  vice  versa  to  a  propeller  too  large  and 
with  a  true  slip  less  than  that  assumed.  If  the  assumed  con- 
ditions correspond  nearly  to  the  maximum  efficiency  as  indi- 
cated by  the  diagram  of  Fig.  71,  then  it  is  probably  better 
that  w  should  be  underrated  rather  than  the  reverse.  This 
is  because  the  efficiency  falls  off  from  the  maximum  much 
more  slowly  for  an  increasing  than  for  a  decreasing  true  slip. 
If,  on  the  other  hand,  the  conditions  fixing  the  efficiency  are 
such  as  to  place  the  propeller  well  over  beyond  the  maximum 
in  the  direction  of  increasing  slip,  then  it  is  better  to  overrate 
the  value  of  w  rather  than  the  reverse.  This  is  because  the 
consequences  will  be  a  decreased  true  slip  and  a  better 
efficiency  than  that  assumed.  Bearing  these  points  in  mind 
we  may  usually,  by  the  aid  of  the  diagram  in  Fig.  71,  take  the 
values  of  w  and  s^  in  such  way  that  the  probable  error  will 
either  increase  the  efficiency  or  insure  the  minimum  amount 
of  decrease.  These  remarks  relate  simply,  of  course,  to  con- 
siderations of  efficiency.  It  will  very  commonly  be  found 
that  the  final  results  adopted  are  largely  dependent  on 
structural  and  other  considerations,  aside  from  efficiency. 

According  to  Blechynden,  whose  experiments  on  pro- 
pellers have  been  mentioned  previously,  the  best  results  will 
generally  be  obtained  from  Froude's  data  by  using  values 
which  would  place  the  propeller  on  the  efficiency  surface  of 
Fig.  71,  somewhat  beyond  the  crest  in  the  direction  of 
increasing  slip.  According  to  the  same  authority  propellers 
of  low  pitch-ratio,  as  for  example  i.i  or  1.15,  are  not  efficient 
when  used  for  the  propulsion  of  ships  of  full  form,  such  as 
cargo-vessels  of  block  coefficient  from  .7  to  .8.  On  thb 


PROPELLER  DESIGN. 


287 


point,  however,  there  is  much  conflicting  testimony,  there 
being  considerable  data  indicating  in  a  general  way  the  suit- 
ability of  moderate  pitch-ratios  for  ships  of  full  form,  and  of 
higher  pitch-ratios  for  ships  of  moderate  and  fine  forms.  The 
question  seems  to  turn  principally  on  the  effect  due  to  the 
augmentation,  this  being  more  pronounced  on  a  full  than  on 
a  fine  after  body.  Now  the  augmentation  increases  in 
general  with  the  diameter  and  with  the  pitch-ratio.  Hence, 
so  far  as  this  effect  is  concerned,  we  should  expect  the  best 
results  from  moderate  pitch-ratio  and  small  diameter,  and 
these  two  conditions,  by  equation  (i),  §  50,  are  seen  to  be 
mutually  consistent.  On  the  other  hand,  with  very  full  ships 
there  will  be  at  the  stern  a  certain  amount  of  dead  or  eddy- 
ing water.  If  the  propeller  is  so  small  and  so  near  the  stern 
as  to  work  for  the  most  part  in  this  eddy,  its  supply  of  water 
will  be  incomplete  and  irregular,  and  the  efficiency  will  be 
correspondingly  decreased.  Judged,  therefore,  from  this 
standpoint  we  should  expect  the  best  results  from  a  propeller 
placed  as  far  aft  as  the  structural  arrangements  will  allow, 
and  of  a  diameter  sufficient  to  reach  out  through  the  eddying 
water  into  that  possessing  regular  stream-line  motion.  Such 
a  value  of  the  diameter  would  naturally  go  with  a  relatively 
large  pitch-ratio.  We  have  therefore  in  this  case,  as  in 
nearly  all  others  with  which  we  have  to  deal,  opposing  con- 
siderations, one  indicating  a  small  diameter  and  the  other  a 
large.  The  actual  best  value  in  any  given  case  will  neces- 
sarily depend  on  the  balance  of  these  considerations,  and  for 
thi-.  in  the  present  state  of  our  information,  no  general  rule 
can  be  framed. 

In    connection    with    the    influence    of    a  very   full    stern, 
reference  may  be  made  to  an  extreme  case,  instances  of  which 


288  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

have  been  said  to  occur.  In  cases  where  the  inflow  of  water 
is  more  or  less  hindered,  the  action  of  the  propeller  may 
approach  more  and  more  nearly  to  that  of  a  centrifugal 
pump,  delivering  the  water  with  a  large  radial  or  transverse 
velocity  and  with  slight  longitudinal  acceleration,  and  thus 
absorbing  energy  with  but  slight  gain  of  thrust.  The  defect 
of  pressure  forward  of  the  propeller  will,  however,  produce  in 
full  measure  its  effect  on  the  hull,  and  in  the  extreme  case 
the  thrust  gained  may  be  less  than  the  sternward  resultant 
due  to  this  defect,  and  thus  the  final  resultant  force  on  the 
ship  may  be  aft  rather  than  forward.  In  such  case  the  ship 
would  move  aft  no  matter  in  what  direction  the  propeller 
turned.  Such  cases  have  been  reported,  although,  of  course, 
they  are  not  possible  with  usual  forms. 

The  Number  of  Propellers. — This  question  is  one  which 
depends  chiefly  on  the  desired  subdivision  of  the  power,  the 
available  draft,  and  the  desired  number  of  revolutions  of  the 
engines. 

In  many  cases  the  total  power  is  more  than  it  seems  desir- 
able to  transmit  with  one  shaft,  and  thus  two  or  more  shafts 
and  propellers  are  fitted.  This  was  the  case  with  the  U.  S. 
triple-screw  cruisers  Columbia  and  Minneapolis,  in  which  it 
seemed  desirable  at  the  time  of  their  design  to  subdivide  the 
total  power  into  three  units  for  transmission  to  the  propellers. 
This  was  only  one  of  several  reasons  for  adopting  this 
arrangement;  but  as  such  it  occupied  a  prominent  place  in 
the.  discussion  of  the  general  problem.  Similar  considera- 
tions hold  for  modern  ocean-liners  developing  from  20,000  to 
30,000  I.H.P.  In  these  cases  it  seems  desirable  to  transmit 
the  power  to  the  propellers  in  two  units,  and  if  there  were  no 
ether  reasons  for  subdivision,  it  is  doubtful  if  under  present 


PROPELLER   DESIGN. 


289 


conditions  engineers  would  wish  to  transmit  the  entire 
amount  by  one  shaft.  Then,  aside  from  reasons  of  this  char- 
acter, we  have  the  more  important  considerations  of  subdivi- 
sion for  safety  and  for  manoeuvring  power.  The  duplication 
of  propelling  machinery  gives  vastly  greater  security  against 
total  breakdown  or  disablement,  and  this,  as  well  as  the  added 
manoeuvring  power  for  war-ships,  furnishes  considerations  of 
sufficient  importance  to  justify  or  rather  demand  the  fitting 
of  twin-screws  in  all  such  cases. 

Again,  the  average  draft  may  be  so  limited  or  trre  power 
so  great  that  a  single  screw  would  be  of  too  large  diameter 
for  the  necessary  immersion  of  its  blades.  This  would  be  the 
case  with  most  of  the  fast  liners  and  war-ships,  and  with  all 
moderately  fast  vessels  of  light  draft.  In  extreme  cases  the 
draft  may  be  so  small  that  three  or  more  propellers  might  be 
needed  in  order  to  absorb  the  necessary  power  with  the 
limited  diameter  permissible.  There  seems  to  be  no  reason,, 
so  far  as  efficiency  of  operation  is  concerned,  why  such  sub- 
division, if  needed,  should  not  be  made.  For  such  cases, 
however,  attention  may  be  called  to  the  peculiar  advantages 
of  the  turbine  propeller  as  described  in  §  40. 

With  regard  to  the  question  of  revolutions,  it  is  obvious 
that  the  smaller  the  propeller,  pitch-ratio  and  speed  being; 
the  same,  the  higher  will  be  the  revolutions.  It  may  arise, 
therefore,  that  the  desired  number  of  revolutions,  if  high,  can 
only  be  attained  with  twin-  or  multiple-screws. 

In  the  question  of  the  applicability  of  triple-screws  to 
cruising  war-ships,  one  of  the  chief  considerations  not  already 
mentioned  related  to  the  resultant  possibility  of  cruising  at 
!<>\v  speeds  with  one  or  two  engines  working  at  nearly  full 
]>'>ucr,  rather  than  with  all  engines  working  at  a  small  frac- 


2QO  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

tion  of  full  power.  In  such  cases  the  thermodynamic  and 
mechanical  efficiencies  should  be  improved.  The  screws  not 
in  operation,  however,  are  dragged,  and  this  increases  the 
resistance.  The  slip  also  usually  increases,  especially  with 
one  screw,  and  this  may  result  in  a  loss  of  propulsive  effi- 
ciency. In  practice,  therefore,  it  has  been  found,  both  in  this 
country  and  in  Europe,  that  the  use  of  one  or  two  screws  at 
low  speeds  does  not  result  in  increased  efficiency.  In  a 
number  of  cases  the  coal  required  for  a  given  speed  with  one, 
two,  and  three  screws  showed  but  slight  variation,  and  the 
trials  seemed  to  indicate  an  approximate  equivalence  in 
general  efficiency.  Notwithstanding  this,  triple-screws  have 
been  extensively  introduced  into  the  practice  of  the  German 
naval  authorities  as  well  as  elsewhere,  the  chief  reasons  being 
the  possibility  of  smaller  and  lower  engines,  smaller  castings 
and  easier  construction,  better  stowage  of  the  engines  in  the 
lean  after  body  of  fast  ships,  and  advantages  arising  in  time 
of  action  from  a  triple  subdivision  of  the  power  and  engine- 
room  personnel. 

In  the  general  question  of  single-  or  multiple-screws  there 
is  probably  no  necessary  difference  in  propeller  efficiency  alone, 
except  such  as  may  arise  from  a  better  possible  selection  of 
characteristics  in  one  case  or  another.  With  the  proper 
estimate  of  wake  and  proper  design  throughout  there  is 
reason  to  believe  that  no  especial  advantage  in  propeller 
efficiency  can  be  expected  on  either  hand,  and  that  the  choice 
must  be  made  rather  upon  the  considerations  discussed  above. 

Location  of  Propellers.  —  The  location  of  a  propeller 
depends  on  the  question  of  augmentation  of  resistance,  on  its 
relation  to  the  wake,  and  on  various  structural  considerations. 
As  already  seen,  the  augmentation  of  resistance  is  more  pro- 


PROPELLER    DESIGN. 


29I 


iiounced  with  a  full  than  with  a  fine  form,  and  with  the 
former  especially  will  vary  in  marked  degree  with  the  dis- 
tance of  the  propeller  from  the  stern-post.  In  such  cases  it 
is  believed  that  the  propeller,  if  single,  should  be  at  a  distance 
from  the  stern-post  not  less  than  one  half  or  two  thirds  its 
diameter.  As  the  distance  is  decreased  from  about  this 
amount  the  flow  of  water  to  the  propeller  becomes  more  and 
more  indirect  and  incomplete,  the  thrust  developed  becomes 
less  than  it  should  be,  and  the  augmentation  effect  becomes 
more  and  more  pronounced.  With  vessels  of  moderate  and 
pronounced  fineness  these  effects  are  less  notable  in  character, 
and  the  propeller  if  single  is  usually  placed  directly  aft  of  the 
^tern-post,  at  such  a  distance  as  the  necessary  structural 
arrangements  make  convenient.  In  torpedo-boats,  launches, 
and  yachts  the  propeller  is  frequently  dropped  below  the  keel 
line  and  is  sometimes  placed  aft  of  the  rudder.  This  favors 
a  full  and  free  flow  of  water  to  the  propeller,  and  tends  to 
decrease  augmentation  of  resistance. 

Twin-screws  are  usually  placed  in  the  same  plane  and 
slightly  forward  of  the  stern-post.  In  such  case  the  average 
di.^tance  from  the  disk  of  the  screw  forward  to  the  ship's  sur- 
face is  much  greater  than  with  a  single-screw,  and  the  aug- 
mentation of  resistance  is  correspondingly  less.  In  some 
cases  an  aperture  has  been  cut  through  the  dead-wood  oppo- 
site the  propellers  in  order  to  give  an  outlet  at  this  point,  and 
thus  relieve  the  stern  from  the  periodic  shock  to  which  it 
might  be  subject  with  the  blades  passing  very  near  the  sur- 
face. In  other  cases  the  screws  have  been  located  in  different 
transverse  planes,  each  with  apertures,  and  with  their  disks 
overlapping.  The  advantages  claimed  are  decreased  distance 
between  the  shafts  with  a  given  diameter  of  propeller,  or 


2Q2  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

greater  diameter  with  given  distance  between  the  shafts, 
seems  doubtful,  however,  whether  the  final  results  are  such 
as  to  recommend  this  arrangement  for  general  adoption. 

With  three  propellers  the  center  one  is  usually  more 
deeply  immersed  than  those  at  the  sides,  the  three  shaft- 
centers  projected  on  a  transverse  plane  forming  a  triangle 
with  apex  downward.  The  center  propeller  is  also  naturally 
farther  aft  than  those  at  the  side,  and  with  this  distribution 
there  seems  to  be  little  interference  between  the  propellers, 
and  no  apparent  loss  of  efficiency  in  action. 

In  general  it  should  be  borne  in  mind  that  the  farther  a 
propeller  is  located  from  the  stern  of  the  ship  the  less  in 
general  will  be  the  resultant  augmentation  of  resistance,  and 
the  less  also  the  valuable  return  from  the  wake.  It  is  evident, 
therefore,  that  so  long  as  the  propeller  is  at  a  sufficient  dis- 
tance from  the  stern  to  obtain  a  good  flow  of  water  and  to 
avoid  an  abnormal  degree  of  augmentation  there  will  be  in 
general  no  advantage  in  its  farther  removal,  for  the  loss  in 
wake  return  will  offset  any  further  gain  by  way  of  a  decrease 
of  augmentation. 


52.    SPECIAL  CONDITIONS  AFFECTING  THE  OPERATION  OF 
SCREW  PROPELLERS. 


Indraught  of  Air. — The  application  of  the  equations  of 
the  preceding  sections  presupposes  that  the  propeller  has 
available  a  full  supply  of  what  is  sometimes  termed  "  solid 
water";  that  is,  water  without  sensible  admixture  of  air.  It 
is  found  as  a  result  of  the  defect  of  pressure  which  exists  in 
the  interior  of  the  stream  flowing  to  and  through  the  pro- 
peller, that  if  the  blades  in  the  upper  part  of  their  path 


PROPELLER    DESIGN. 


293 


approach  near  the  surface  of  the  water,  and  more  especially  if 
they  cut  or  break  the  surface,  that  air  is  drawn  down  and 
mixed  with  the  water.  The  result  is  a  more  or  less  frothy  or 
foamy  mixture,  the  density  of  which  is  much  less  than  that  of 
water.  In  consequence  the  thrust  for  given  revolutions  and 
slip  is  decreased,  or  for  given  thrust  the  revolutions  and  slip 
mu-t  be  increased.  In  either  case  the  efficiency  is  generally 
decreased.  If  the  tips  of  the  blades  come  very  near  the  sur- 
face or  actually  cut  it,  the  loss  in  thrust  and  efficiency  may 
become  serious,  especially  with  large  pitch-ratio  and  high 
peripheral  velocity. 

It  should  be  clearly  understood  that  a  loss  of  efficiency  is 
by  no  means  necessarily  attendant  on  the  working  of  a  pro- 
peller in  foam,  or  on  the  mere  fact  that  the  tips  of  the  blades 
come  near  the  surface  or  actually  cut  it.  The  whole  ques- 
tion, so  far  as  the  operation  of  the  propeller  is  concerned,  is 
determined  by  the  resultant  true  slip  at  which  it  works.  It 
would  be  perfectly  possible  to  design  a  propeller  with  blades 
cutting  the  surface  but  with  sufficient  diameter  so  that  even 
with  the  indraught  and  admixture  of  air  the  true  slip  would 
be  within  the  proper  range  for  good  efficiency. 

In  order,  however,  to  keep  down  the  size  of  the  propeller 
and  to  insure  its  working  in  water  prac'.ically  free  from  air,  its 
diameter  should  be  so  chosen  with  reference  to  the  draft  of 
water  at  the  stern,  as  modified  by  the  possible  change  of 
trim  and  stern-wave,  that  the  tips  of  the  blades  will  always 
be  well  covered.  Experience  seems  to  indicate  that  the 
desirable  immersion  of  the  tips  of  the  blades  when  nearest 
the  surface  should  not  be  less  than  about  .2  or  .3  the 
diameter  of  the  propeller.  In  addition  to  the  indraught  of 
air  from  the  surface  there  is  always  under  normal  conditions 


294  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

a  certain  amount  of  air  held  in  absorption  by  the  water*. 
This  air  is  given  off  more  or  less  completely  about  the  edges 
and  backs  of  the  blades  in  consequence  of  the  partial  vacua 
which  exist  at  these  points.  This  will  give  rise  unavoidably 
to  a  small  amount  of  foam  in  the  wake,  the  consequences  of 
which,  however,  are  believed  to  be  insensibly  small. 

Very  Full  Form  at  the  Stern. — Mention  has  been  made  of 
this  feature  in  the  preceding  section. 

Racing. — When  a  ship  is  in  a  seaway  the  blades  are  con- 
tinually varying  their  distances  from  the  surface  or  actually- 
emerging  from  it,  and  the  revolutions  undergo  correspond- 
ingly sudden  and  perhaps  extreme  variations,  unless  by  some 
form  of  governing  device  the  supply  of  steam  to  the  engine 
is  suitably  controlled.  Even  at  the  best  the  revolutions,  and 
hence  the  slip,  will  frequently  vary  through  very  wide  ranges. 
Such  variations  may  also  be  due  partly  to  the  effect  of  the 
internal  motion  of  the  water  composing  a  wave,  the  move- 
ment being  forward  in  the  crest  and  aft  in  the  hollow.  Such 
variations  in  the  speed  of  the  propeller  are  favorable  to  the 
production  of  eddies,  and  thus  to  a  waste  of  energy  and  loss 
of  efficiency.  There  is  in  addition  a  further  loss  in  efficiency 
due  to  the  wide  range  of  slip  through  which  the  propeller 
.works.  We  know,  as  indicated  by  Fig.  71,  that  for  the  best 
results  the  propeller  should  work  at  or  close  about  a  single 
value  of  the  slip.  But  in  violent  racing  the  slip  for  the  pro- 
peller as  a  whole  may  vary  from  perhaps  a  negative  value  to 
50  or  75  per  cent  or  even  more.  Having  reference  to  Fig. 
71  it  is  readily  seen  that  such  wide  variation  in  the  value  of 
the  slip  must  inevitably  cause  a  serious  falling  off  in  the  aver- 
age efficiency.  Other  things  being  equal,  these  effects  will 
be  more  pronounced  the  more  readily  the  propeller-blades  are- 


PROPELLER    DESIGN. 


-95 


partially  uncovered,  and  hence  the  nearer  they  are  to  the  sur- 
face of  the  water  in  normal  trim.  We  have  here,  therefore, 
an  additional  reason  for  good  immersion  of  the  propellers.  It 
is  here  that  twin-screws  gain  one  of  their  advantages,  their 
diameter  being  less  and  possible  immersion  greater  than  for  a 
single-screw  to  absorb  the  same  total  power. 

Capitation  or  the  Effect  due  to  Very  High  Propeller  Veloci- 
ties. — If  a  blade  such  as  AB,  Fig.  83,  is  drawn  through  the 
A 


FIG.  83. 

water  as  indicated,  the  space  immediately  in  its  rear  is  ren 
dered  more  or  less  empty  on  account  of  the  time  required  for 
the  water  to  close  in  behind  the  moving  blade.  At  slow 
speeds  this  open  space  will  only  extend  slightly  below  the 
surface  of  the  water,  while  if  the  speed  is  sufficiently  great  it 
may  reach  to  the  bottom  of  the  blade,  being  shaped  in  con- 
tour somewhat  as  shown  by  BCD.  The  velocity  in  feet  per 
second  with  which  water  tends  to  flow  into  an  open  space  is 
given  by  the  general  formula  v  —  V2g/i.  Where  the  space 
is  open  to  the  air,  as  in  Fig.  83,  h  =  simply  the  head  of 
water.  Thus,  for  example,  at  the  bottom  of  a  board  immersed 
depth  of  4  feet  v  =  16,  while  if  the  depth  is  I  foot 
i>  =  8.  Where  the  space  is  shut  off  from  the  air  and  is  a 
perfect  vacuum,  //  =  the  head  due  to  the  atmospheric  pressure 
plus  that  clue  to  the  water,  or  //  =  34  feet  plus  depth  of 


296  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

water.  In  any  actual  case  the  vacuum  cannot  be  perfect,  but 
will  contain  water-vapor  and  some  air  given  off  from  the 
water.  In  consequence  the  values  for  such  case  will  be  less 
than  those  resulting  from  the  head  given  above,  which  must 
be  considered  rather  as  an  upper  or  limiting  value. 

Now  if  the  velocity  of  the  blades  of  a  propeller  is  suffi- 
ciently high  it  is  evident  that  such  spaces  will  be  formed  at 
the  back — in  this  case  of  course  entirely  under  water.  When 
this  phenomenon  is  present  in  any  marked  degree  the  water 
acted  on  by  the  propeller  is  found  to  become  turbulent 
instead  of  measurably  continuous,  and  with  further  increase 
of  revolutions  the  gain  in  thrust  is  very  slight. 

This  phenomenon  has  been  investigated  by  M.  Normand* 
for  a  torpedo-boat  fixed  in  location  instead  of  in  free  route. 
The  diameter  of  propeller  was  about  6.5  feet  and  mean  pitch 
7.7  feet.  The  center  of  the  shaft  was  maintained  at  four 
immersions  varying  from  4.8  to  3.85  feet.  The  curves  con- 
necting thrust  with  revolutions  are  as  shown  in  Fig.  84,  in 
•which  the  ordinate  is  the  square  of  the  revolutions  N,  and 
the  abscissa  is  the  resultant  thrust.  The  regular  increase  of 
thrust  with  N*  for  the  deepest  immersion  is  shown  by  the 
approximately  straight  line  OD.  For  the  lightest  immersion 
and  moderate  revolutions  the  thrust  was  slightly  greater  than 
for  the  deepest  immersion,  but  the  difference  was  not  large, 
and  the  increase  with  the  square  of  the  revolutions  was 
regular.  After  reaching  about  135  revolutions,  however,  the 
rate  of  increase  of  thrust  with  revolutions  began  quickly  to 
decrease;  and  from  this  point  on  there  was  but  slight  increase 
of  thrust  for  increase  of  revolutions  up  to  about  230,  the 


*  Bulletin  de  1'Association  Technique  Maritime,  vol.  iv.  p.  68. 


PROPELLER   DESIGN. 


297 


maximum  number  obtained.  Very  near  the  end,  however, 
as  shown  by  the  curve,  the  rate  of  increase  was  slightly 
greater  than  at  intermediate  points.  M.  Normand  suggests 
that  the  curve  between  B  and  C  corresponds  to  a  period  of 


1000 


2000       3000        4000        5000        6000 
THRUST  IN  KILOGRAMMES. 

Fir,.  84. 


7000 


instability  of  the  open  spaces,  during  which  they  are  being 
formed  and  again  filled  with  more  or  less  irregularity,  thus' 
increasing  the  turbulence  of  the  water,  while  beyond  C  the 
condition  is  more  permanent,  and  the  increase  of  thrust  with 
N*  is  somewhat  more  rapid.  The  remaining  curves  give 
similarly  the  thrusts  for  intermediate  immersions. 

The  direct   application  of  this  to  the  case  of  vessels  in 

route  is  somewhat  uncertain,  because  the  conditions  are 

quite  different.      The  slip  is  much   less,  the  amount  of  water 

handled   much   more,   and   the  actual  velocity  of  the  blades 


298  RESISTANCE   AND   PROPULSION  OF  SHIPS. 

through  the  water  greater  for  the  same  revolutions.  It 
seems  quite  certain  that  in  free  route  the  revolutions  might 
be  considerably  greater  without  rupture  of  the  column  than 
for  a  boat  fixed  in  location. 

Still  more  recently  these  phenomena  have  been  independ- 
ently examined  by  Parsons  ~x'  and  Barnaby  f  in  England. 
According  to  the  latter,  the  limiting  conditions  when  cavita- 
tion  is  about  to  set  in  are  more  readily  expressed  in  terms  of 
the  average  thrust  developed  per  unit  of  projected  blade  area 
than  in  terms  of  velocity,  and  both  investigations  furnished 
for  this  limiting  thrust  substantially  the  same  value  of  nj 
pounds  per  square  inch,  the  immersion  of  the  tips  of  the 
blades  being  1 1  inches.  At  the  point  where  this  average 
thrust  per  square  inch  of  projected  blade  area  was  reached 
signs  of  cavitation  began  to  appear,  and  on  increasing  the 
revolutions  the  phenomenon  became  more  or  less  pronounced, 
and  but  little  additional  thrust  was  gained.  Barnaby  states 
also  that  for  every  additional  foot  of  immersion  the  thrust  per 
square  inch  may  be  increased  about  -f  pound.  The  value 
should  also  vary  slightly  with  pitch-ratio,  being  less  if  the 
latter  is  large ;  but  the  influence  due  to  this  feature  is  so  small 
as  to  be  practically  negligible. 

It  follows,  therefore,  according  to  these  investigations, 
that  from  11  to  12  pounds  per  square  inch  of  projected  blade 
area  may  be  considered  as  the  maximum  thrust  which  can  be 
efficiently  developed,  and  if  more  than  this  is  required  it  can 
only  be  attained  with  difficulty  and  at  a  loss  of  efficiency. 
The  phenomenon  of  cavitation  has  presented  itself  more  or 
less  prominently  in  connection  with  the  modern  torpedo- 


*  Transactions  Institute  of  Naval  Architects,  vol.  xxxvin. 
f  Ibid. 


PROPELLER   DESIGN. 


299 


boat  destroyer  and  boats  of  similar  type  or  condition.  In 
such  cases  2500  to  3500  I.H.P.  provided  for  each  of  two  pro- 
pellers 6  to  7  feet  in  diameter  at  400  revolutions  and  upward 
develop  a  speed  of  30  knots  and  upward.  It  seems  probable 
that  this  phenomenon  may  cause  trouble  with  further  increase 
of  speed,  especially  in  boats  of  this  character;  and  the  all- 
important  question  in  such  cases  is  therefore  as  to  the  best 
disposition  of  proportions  and  dimensions  for  the  develop- 
ment of  the  maximum  thrust  per  unit  area  without  undue 
sacrifice  of  efficiency  due  to  this  cause.  The  obvious  way  to 
avoid  the  trouble,  where  possible,  is  to  provide  a  sufficient 
projected  area  of  blade  to  reduce  the  average  thrust  below 
the  limiting  value,  and  the  use  of  nine  propellers  on  the 
Turbinia  is  an  illustration  of  this  treatment  of  the  difficulty. 

Springing  of  Propeller-blades. — From  the  laws  regulating 
the  distribution  of  pressure  over  a  plane  moving  through  'the 
water  as  explained  in  §  6,  it  follows  that  the  outer  portions 
of  propeller- blades,  or  such  portions  as  may  be  considered 
substantially  equivalent  to  thin  flat  surfaces,  will  tend  to  place 
themselves  normal  to  the  stream  or  line  of  flow.  This  would 
result  in  a  tendency  to  increase 
the  pitch  and  slip.  The  natural 
tendency  of  the  inner  portions  or 
those  with  a  sensibly  rounded 
back  will  be  quite  different.  It  is 
readily  shown  by  experiment  that 
when  placed  slightly  oblique  the 
tendency  of  such  sections  is  indeed 
to  place  themselves  at  right  angles  to  the  stream,  but,  as 
illustrated  in  Fig.  85,  the  incipient  movement  is  clockwise 
instead  of  the  reverse,  the  final  position  toward  which  the 


FIG.  85. 


300  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

section  tends  being  as  shown  in  dotted  lines,  with  convex 
side  toward  the  stream.  This  will  result  in  a  tendency  to 
decrease  the  pitch  and  slip.  (Compare  also  §  42,  Fig.  64.) 

In  addition  to  the  tendency  to  spring  due  to  the  irregu- 
larity of  the  distribution  of  pressure,  the  blade  as  a  whole 
will  bend  more  or  less  under  the  influence  of  the  thrust. 
This  bending  will  be  accompanied  by  a  slight  untwisting  of 
the  blade,  thus  tending  toward  an  increase  of  pitch  and  slip. 
This  influence  is  relatively  more  important  than  the  others 
mentioned  above,  and  hence  in  the  complex  effect  the  result 
is  usually  a  slight  untwisting  of  the  blade  with  increase  of 
pitch  and  slip.  With  blades  of  the  usual  stiffness  it  does  not 
seem  likely  that  this  effect  will  be  sufficient  to  produce  any 
sensible  influence  on  the  performance  of  the  propeller.  In 
extreme  cases,  however,  where  the  blades  have  been  thinned 
beyond  the  proper  limit,  the  effective  pitch  and  slip  might 
be  so  increased  as  to  sensibly  affect  the  efficiency — usually 
for  the  worse. 

53.    THE   DIRECT  APPLICATION  OF    THE   LAW   OF  COM- 
PARISON TO  PROPELLER  DESIGN. 

Instead  of  the  method  of  propeller  design  indicated  in  the 
preceding  sections,  it  is  possible  to  apply  directly  the  laws  of 
comparison  as  explained  in  §  26.  As  required  by  the  nature 
of  the  law,  we  assume: 

(1)  Similarity  in   geometrical   form  and  situation  with  a 
linear   ratio   A,   the   same  as  that  which  exists  between   the 
ships  themselves. 

(2)  Corresponding  speeds  as  defined  for  ships,  or  linear 
speeds  in  the  ratio  of  A*.     This  condition  will  evidently  apply 


PROPELLED   DESIGN. 


3O1 


only  to  corresponding  points  on  the  two  propellers.  Let  rl 
and  r^  be  the  radii  of  such  points  on  the  two  propellers,  and 
Nlt  Nt  the  revolutions.  Then  2nrlNl  and  2^r,7Vra  will  be  the 
actual  or  linear  speeds  of  these  points.  Hence  we  shall  have 
for  corresponding  speeds 


%%  _ 
~ 


But 


-' 
r, 


Hence      w  =  AT 


Hence  at  corresponding  speeds  the  revolutions  will  be 
inversely  as  the  linear  speeds,  or  inversely  as  the  square  root 
of  the  linear  dimension-ratio. 

It  must  be  remembered,  further,  that  these  assumptions 
implicitly  involve  equal  percentages  of  slip,  equal  values  of 
the  wake  factor,  equal  values  of  the  factor  of  augmentation 
due  to  the  propeller,  and  equal  mechanical  efficiencies  for  the 
engines. 

It  then  results,  on  the  assumptions  of  §  26,  that  the  ratio 
between  the  powers  necessary  to  drive  the  two  ships  is  A*. 
With  the  propellers  thus  assumed,  it  is  seen  that  they  fulfil 
in  all  points  the  conditions  of  the  law  of  comparison,  consider- 
ing it  applicable  to  the  whole  resistance.  Hence  the  ratio  of 
the  resistances  to  transverse  motion,  the  ratio  of  the  thrusts, 
and  in  fact  the  ratio  of  all  similar  components  of  the  forces 
acting  on  them,  will  be  A1.  The  speed-ratio  is  A*.  Hence  the 
ratio  between  the  amounts  of  useful  work  will  be  A%  the 
same  as  that  for  the  total  amounts;  or,  in  other  words,  the 


302  RESISTANCE   AND  PROPULSION   OF  SHIPS. 

two   propellers    thus  situated  will    give    similar   results  with 
equal  efficiencies. 

We  have  therefore 


A" 


For  example,  let  A  ==  1.44,  ^  =  16  knots,  A^,  =  100. 
Then  A*  =  1.2  .  A3  =  displacement-ratio  =  2.986,  and  A*  = 
power-ratio  =  3.58.  Then  u^  =  19.2  and  Nz  —  83.3.  It 
follows,  therefore,  that  the  larger  ship  would  require  a  propeller 
1.2  times  the  size  for  the  smaller,  and  with  an  expenditure  of 
3.58  times  the  power  at  83.3  revolutions  would  drive  it  at  a 
speed  of  19.2  knots,  with  the  same  efficiency  as  for  the 
smaller. 

This  method  by  comparison  is  really  a  special  case  of  that 
discussed  in  the  preceding  section.  The  method  by  com- 
parison is  a  restricted  mode  of  applying  the  results  of  a  given 
propeller  to  the  design  of  one  of  similar  form  under  similar 
conditions  and  at  corresponding  speeds,  while  the  method  of 
§  50  assumes  such  laws  and  relations  as  will  provide  for  the 
application  of  the  results  of  a  given  propeller  to  the  design 
of  one  of  similar  form  under  similar  conditions  but  at  any 
speeds.  The  relation  between  §  50,  (i),  and  the  law  of  com- 
parison is  readily  seen  by  applying  the  former  to  the  design 
of  a  propeller  using  the  data  of  a  similar  case  at  correspond- 
ing speed.  Taking  first  the  latter,  we  have 


PROPELLER   DESIGN. 

We  suppose  in  this  equation  U,  d,  p,  and  N  as  well  as  the 
speed  and  all  characteristics  of  the  propeller  to  be  the  given 
data.  Then  the  resulting  combined  factor  (ikltii)  is  readily 
found.  We  next  propose,  by  means  of  this  general  equation, 
to  use  this  data  for  the  design  of  a  propeller  for  a  similar  ship 
at  corresponding  speed.  Assuming  the  same  slip,  pitch-ratio, 
and  other  proportions  for  the  propeller,  the  factor  (Mm)  will 
be  the  same  in  both  cases.  £/3  from  the  law  of  comparison 
for  power  will  be  A*  £7,  ,  and  since  the  speeds  are  correspond- 
ing, (p'Nf\  will  equal  \*(p'N'\.  Hence  for  the  second  case 
we  shall  have 

Altfi  =  A2(/A^X''(^«)i  ....     (2) 

Comparing  this  with  (i),  it  is  evident  that 


or 
whence 

and 


=  AW/' 


/>,  =  A/,, 


N,  =    - 


Hence  the  propeller  will  have  the  same  factor  of  similitude 
as  the  ship,  and  will  have  its  revolutions  in  the  inverse  ratio  of 
the  square  root  of  this  factor,  the  same  as  if  determined  by 
the  law  of  comparison  discussed  above.  In  other  words,  the 
application  of  §  50,  (i),  to  such  a  case  in  the  manner  above 
shown  would  give  exactly  the  same  results  as  would  be  given 
by  the  direct  application  of  the  law  of  comparison  itself. 


54.  THE  STRENGTH  OF  PROPELLER-BLADES. 

The   blade  of  a  propeller  may  be  considered   as  a  beam 
supported   at   one  end,  the  root,  and  loaded  in  a  variable  and 


304  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

complex  manner  according  to  the  distribution  of  pressure 
over  its  surface.  As  in  §  35,  the  amount  of  this  pressure  may 
be  approximated  to,  and  by  the  proper  treatment  as  explained 
in  mechanics,  and  by  approximate  integration,  the  amount 
of  the  resultant  bending  moment  at  the  root  may  be  de- 
termined.* 

In  such  case,  however,  we  must  not  forget  that  the  as- 
sumptions are  necessarily  somewhat  inexact,  that  the  actual 
pressure  will  undergo  wide  variations  due  to  irregularities  in 
the  wake,  and  that  as  in  all  structural  design  a  considerable 
factor  of  safety  must  be  introduced  in  order  to  allow  for  the 
unknown  and  uncertain  elements  necessarily  involved  in  the 
operation  of  the  propeller.  We  propose,  therefore,  to  sub- 
stitute for  the  actual  blade  a  rectangular  plane  area  set  at  an 
angle  a  to  the  transverse.  The  value  of  this  angle  will  be 
considered  at  a  later  point. 

Let  n  be  the  number  of  blades,  and  T  the  actual  thrust. 
Then  T -r-  n  —  thrust  for  one  blade,  and  (T  -f-  ri)  sec  a  will 
be  the  normal  pressure  on  the  blade.  The  center  of  this 
pressure  will  correspond  to  the  center  of  a  system  of  forces 
varying  as  the  square  of  their  distance  from  the  axis.  Hence 
denoting  the  outer  radius  and  that  of  the  hub  by  r,  and  r^  we 
have,  as  the  distance  from  the  axis  to  this  center  of  pressure, 


~4'r,'-rt>- 

If  we  take  approximately  r^  =  .2r,,  we  find 
r  =  .756^,     or  sensibly     -7Sri- 
We  shall  also  omit  all  account  of  bending  moment 
to  the  tangential  forces,  as  they  will  in  any  case  be  small 

*  See  Taylor,  "  Resistance  of  Ships  and  Screw  Propulsion,"  p.  208. 


PROPELLER   DESIGN. 


305 


amount,  and  will  be  sufficiently  provided  for  by  the  factor  of 
safety,  and  by  the  reference  of  our  final  equation  to  actual 
results  for  the  determination  of  the  value  of  the  empirical 
factor  which  is  to  be  introduced.  The  bending  moment  at 
the  root  reckoned  normal  to  the  face  is  therefore 


M=  (.75^  — 


T 

-  sec  a. 

n 


Now  let  b  denote  the  length  of  the  section  where  the 
blade  joins  the  hub,  and  /  the  maximum  thickness.  We  may 
safely  consider  that  the  geometrical  characteristics  of  this 
section  will  not  widely  vary.  In  actual  form  it  closely  re- 
sembles a  circular  or  parabolic  segment.  The  general  equation 
for  the  beam  gives  us 

'         KI 
M=  — , 

Jo 

where  K  =  stress  at  outer  fiber ; 

y^  =  distance  from  neutral  axis  of  section  to  such  fiber; 
/  =  moment  of  inertia  of  section  about  neutral  axis. 

With  the  suppositions  above  made  regarding  the  nature  o£ 
the  section  we  should  have 


-  proportional  to  b? . 

y* 


Hence  we  may  put 


I_ 

y* 


and 


(•75*-,  -  ',)-  sec  «  = 


•     •     •     (3) 


3O6  RESISTANCE  AND   PROPULSION   OF  SHIPS. 

Now,  turning  to  Fig.  66,  we  readily  see  that 


g(LH.P.)  X  33QQQ 


(i  -  s 

where  a  represents  the  ratio  (Thrust  H.P.)  -=-  (I.H.P.). 

For  OL  we  shall  take  the  inclination  at  the  assumed  center 
of  pressure,  or  rather  at  ,jrl  in  order  to  allow  somewhat  for 
the  rounding  off  of  the  blades  at  the  outer  end.  We  have, 
then, 

P  P 


2nrl 
Then  if  p  -i-  d  =  c  as  heretofore,  we  have 


c  A/  c 

tan  a  = and    sec  a  =  \  I  -I — . 

2.2  4.84 

Collecting  and  substituting  in  (3)  we  have,  finally, 


3300Q(.75r.  ~  r,XI.H.P.)  Vi  +  t  -4-  4.84        ,. 
(i  -  s^pNKQn 

This  may  be  further  simplified  by  making  the  following 
assumptions,  which  are  quite  admissible,  considering  the  un- 
certainties necessarily  involved : 

r^  =  .2rl  ,     as  above, 

(i  —  ^,)  and  a  sensibly  constant. 

Then  uniting  all  constants  in  one,  which  we  denote  by  A*, 
we  have 


Also,  let  us  put  '-          =  e  and  I. H.P.  =  H. 


PROPELLER   DESIGN. 


307 


Then 


Nn  ' 


or 


bNn 


(7) 


(8) 


In  this  it  must  be  remembered  that  A  includes  the 
strength  of  the  material,  and  hence  its  value  must  be  taken 
accordingly.  This  result,  it  must  be  remembered,  gives  the 
thickness  at  the  actual  root  of  the  blade.  The  fillet  by  which 
it  is  connected  to  the  hub  is  of  course  extra,  and  is  not  here 
considered.  The  values  of  e  may  be  taken  from  the  following 
table: 


=     -f-  d 


e 
i     i.io 

I.I 1.02 

1.2 95 

1.3-.. 89 

1.4 85 

1.5 81 

i. 6..  .77 


1.7.  . 

d 

1.8   . 

I.Q    . 

y     ' 
2.O.  . 

2.1  .. 

2.2  .  . 

2.3     . 

e 

-74 
.72 
.70 
.68 
.66 
.64 
•63 


Now  comparing  this  equation  with  a  large  number  of 
propellers  which  have  shown  sufficient  strength,  it  is  found 
that  for  bronze  or  steel  A  may  be  taken  at  from  9  to  12,  while 
for  cast  iron  its  value  should  be  increased  to  from  14  to  17. 

Collecting  for  convenience  these  various  items,  we  have 
therefore 


t- 


bN~n' 


308  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

where  t  =  thickness  in  inches  at  root  of  blade ; 
H=  I.H.P.; 

e  —  factor  taken  from  the  table  above ; 
b  =.  length  in  inches  of  section  at  root  of  blade ; 
N  =  revolutions  per  min.  ; 
n  =  number  of  blades; 

9  to  12  for  bronze  or  steel; 


A         _ 

^  14  to  17  for  cast  iron. 


MATERIALS    SUITABLE   FOR   SCREW-PROPELLERS. 

Though  the  fundamental  purpose  in  the  present  volume  is 
the  design  of  the  dimensions  and  form  of  propellers,  yet  some 
brief  notice  of  the  materials  suitable  may  not  be  out  of  place. 
Cast  iron,  cast  steel,  brass,  gun-metal,  and  the  various 
bronzes  are  used.  Cast  iron  is  the  cheapest,  but  being  rela- 
tively weak  and  brittle  the  blades  are  necessarily  thicker  and 
less  efficient  than  with  steel  or  bronze.  The  strength  avail- 
able is  usually  from  2OOOO  to  25  ooo  Ibs.  per  square  inch  of 
section.  Cast-iron  blades  may  be  broken  short  off  by  striking 
logs  or  obstacles,  and  if  such  collisions  are  likely  to  occur  it 
might  be  better  to  have  cast-iron  blades  than  steel  or  bronze, 
as  the  rupture  of  a  blade  might  spare  more  serious  results 
arising  from  strains  on  the  engine  and  stern  of  the  ship. 
Usually,  however,  we  wish  the  blades  to  hold  and  not  to 
break,  and  for  this  purpose  cast  iron  is  at  a  relative  disad- 
vantage. 

Cast  steel  is  stronger  than  cast  iron,  its  ultimate  strength 
in  castings  suitable  for  propeller-blades  ranging  from  50  ooo 
to  60  ooo  Ibs.  per  square  inch  of  section.  The  sections  may 
therefore  be  made  thinner,  and  a  better  efficiency  obtained  in 
so  far  as  dependent  on  this  feature.  The  surface  is  naturally 


PROPELLER   DESIGN. 


309 


not  as  smooth  as  that  of  cast  iron,  but  with  improved 
methods  of  production  the  difference  in  this  feature  is  insig- 
nificant. 

Bronzes  have  naturally  a  smoother  surface,  and  seem, 
furthermore,  to  have  a  lower  coefficient  of  skin-resistance. 
This  added  to  their  strength  and  good  casting  qualities  makes 
possible  a  relatively  smooth  thin  blade  with  sharp  edges,  all 
of  which  are  features  favorable  to  good  efficiency.  The 
strength  available  with  the  best  bronzes  varies  from  40  ooo  to 
60  ooo  Ibs.  per  square  inch  of  section.  With  ordinary  gun- 
metal  from  25  ooo  to  35  ooo  Ibs.  per  square  inch  of  section 
may  be  allowed,  while  with  common  brass  not  more  than 
20  ooo  to  25000  Ibs.  should  be  depended  on.  Of  these 
various  alloys,  manganese-bronze  is  probably  more  used  than 
any  other,  due  to  its  better  combination  of  desirable  qualities 
such  as  strength  and  stiffness,  good  casting  qualities,  resist- 
ance to  corrosion,  etc.  Care  is  needed  in  the  manipulation  of 
the  bronzes  in  melting,  pouring  and  cooling  in  order  to  obtain 
the  full  benefit,  but  with  such  care  the  product  is  homogene- 
ous and  reliable.  Its  greater  relative  cost  restricts  its  use, 
however,  to  war-ships,  yachts,  and  launches,  ocean-liners,  and 
other  cases  where  the  importance  of  a  saving  in  propulsive 
efficiency  is  considered  worth  obtaining  at  a  slight  increase  in 
first  cost. 

The  durability  of  propeller-blades  is  in  the  order:  bronze, 
cast  iron,  cast  steel.  The  two  latter  usually  deteriorate  by 
general  corrosion  and  local  pitting,  the  average  life  being 
usually  from  five  to  ten  years.  The  life  of  bronze  blades  is 
practically  indefinite,  or  at  least  as  great  as  that  of  the  ship 
Itself. 


3IO  RESISTANCE  AND    PROPULSION   OF  SHIPS. 


55.  GEOMETRY  OF  THE  SCREW  PROPELLER. 


Some  general  notions  relating  to  the  geometrical  form  of 
the  screw  propeller  have  been  given  in  §  34.  We  have  now 
to  give  a  more  detailed  account  of  the  principal  forms  which 
may  be  produced. 

We  may  define  the  surface  of  a  screw  propeller  in  general 
as  a  surface  generated  by  a  line  /  moving  on  two  other  lines  as 
guides,  of  which  one,  a,  is  straight,  forming  the  axis,  and  the 
other,  #,  lies  in  the  surface  of  a  cylinder  of  which  a  is  the  axis. 
This  surface  being  developed,  we  have  the  actual  form  of  b. 
For  the  common  uniform-pitch  propeller  with  blades  at  right 
angles  to  the  axis  it  is  readily  seen  that  /  is  straight  and  at 
right  angles  to  a,  and  that  the  developed  b  is  straight  and  at 
an  angle  a  with  the  transverse  plane  such  that  we  shall  have 


pitch  —  2nr  tan  a, 


where  r  is  the  radius  of  the  cylinder  in  which  b  is  supposed 
to  lie. 

The  line  /  is  the  generatrix ; 
'  *       '  '   a  *  *    '  *    axis ; 
11       "   b  "    "    guide,  or  guide-iron. 
We  will  denote  b  when  developed  by  b. 

The  variations  usually  found  in  /  are  as  follows : 

(1)  Straight  and  at  right  angles  to  a\ 

(2)  "          "      inclined  to  a\ 

(3)  Bent  or  curved  in  an  axial  plane ; 

(4)  "      "        ll       "a  transverse  plane. 
The  axis  a  is-  invariable  in  character. 


PROPELLER   DESIGN. 

The  guide  b  may  vary  as  follows : 

(1)  Straight; 

(2)  Curved. 

The  curvature  under  (2)  is  usually  such  that  the  angle  at 
and  hence  the  pitch,  increases  from  the  forward  to  the  after 
edge  of  the  blade. 

In  the  actual  generation  of  the  surface  we  may  also  have 
variations  according  as  to  whether  /  changes  its  angular  position 
relative  to  a  or  not.  If  the  longitudinal  component  of  the 
motion  of  the  point  of  contact  of  /  with  b  is  greater  or  less  than 
that  of  the  point  of  contact  of  /  with  a,  the  pitch  will  vary 
radially,  or  from  the  hub  outward.  In  such  a  case  practically 
two  guides  would  be  required — one  near  the  hub  to  suit  the 
pitch  at  that  radius,  and  the  other  as  usual  at  the  outer  end 
and  suited  to  the  pitch  at  that  point. 

Let  c  denote  the  ratio  of  the  velocities  of  the  points  of 
contact  of  /  with  a  and  with  b.  Then  we  may  have  three 
cases: 

(1)  c  equal  to  I  ; 

(2)  c  greater  than  I  ; 

(3)  c  less  than  i. 

All  common  varieties  of  helicoidal  surfaces  may  be  gener- 
ated by  giving  to  /,  b,  and  c  the  different  variations  as  above 
enumerated.  We  may  have,  however,  other  forms  of  heli- 
coidal surface  which  cannot  be  generated  by  a  rigid  line  in  any 
of  the  ways  above  mentioned.  We  may  most  readily  conceive 
of  such  a  surface  as  made  up  of  the  summation  of  an  indefinite 
number  of  guides  b,  and  thus  as  generated  by  a  variable  guide 
moving  radially,  and  perhaps  angularly  and  longitudinally  as 
well,  and  taking  at  the  same  time  the  successive  shapes  and 
inclinations  as  determined  by  the  nature  of  the  pitch  at  the 


312  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

successive  radial  locations.  In  this  way  any  form  of  blade 
and  any  distribution  of  pitch,  no  matter  how  complex,  may 
be  geometrically  determined,  and  by  the  use  of  a  reasonable 
number  of  guides  actually  produced  in  the  form  of  a  pattern, 
or  moulded  direct  in  the  foundry. 

Usually,  however,  the  surfaces  of  propellers  are  such  as 
can  be  generated  by  a  rigid  line,  and  of  the  great  variety 
which  may  thus  be  formed  but  few  are  commonly  found, 
These  are  as  follows : 

l^blcl  gives  the  common  propeller  of  uniform  pitch ; 

IJ)^  gives  the  propeller  of  uniform  pitch  with  straight  ele- 
ments bent  back  from  the  plane  of  revolution ; 

IJ>\C\  gives  the  propeller  of  uniform  pitch  as  in  the  last,  but 
with  the  elements  curved  away  from  the  plane  of  revo- 
lution ; 

IJ*\C\  gives  the  propeller  of  uniform  pitch  with  elements  bent 
or  curved  in  the  plane  of  revolution ; 

llbtcl  gives  the  common  expanding-pitch  propeller,  the  pitch 
increasing  from  forward  to  after  edge. 


We  may  also  form  these  various  combinations  with  c^  or  cs 
and   thus    introduce    radial    variation    of   pitch    in    any    way 
desired. 

It  is  also  possible  to  greatly  vary  the  appearance  of  a  pro- 
peller-blade by  varying  the  location  of  the  part  actually  taken 
for  the  blade.  Thus  from  a  helicoidal  surface  of  uniform 
pitch,  by  appropriately  locating  and  shaping  the  contour,  we 
may  produce  a  blade  which  has  the  appearance  of  being  bent 
back  from  the  plane  of  revolution,  and  which  on  casual  exam- 
ination may  seem  to  be  the  same  as  that  given  by  ljflcl  or 
lj)j^  though  such  is  by  no  means  the  case. 


LX 

: 


PROPELLER   DESIGN. 


313 


While  thus  an  indefinite  variety  of  propellers  are  possible 
as  regards  shape  of  blade  and  distribution  of  pitch,  it  must  be 
clearly  understood  that  so  far  as  experimental  or  actual  results 
are  concerned  we  are  not  in  a  position  to  select  among  them 
any  one  set  of  characteristics  as  the  best,  and  especially  any 
one  distribution  of  pitch,  or  indeed  to  say  that  any  distribu- 
tion of  pitch  wil1  produce  a  propeller  superior  to  that  of  uni- 
form pitch.  It  is  possible  that  some  distribution  may  be 
superior  to  the  uniform,  but  we  are  not  at  present  in  a  posi- 
tion to  either  confirm  or  disprove  this  by  experimental  data. 
We  find  therefore  a  very  general  tendency  to  consider  the 
propeller  of  uniform  pitch  as  the  standard  in  type.  The  bent- 
back  blades  as  given  by  /,£,<:,  has  been  much  used  on  tor- 
pedo-boats, yachts,  and  small  craft,  the  presumable  reasons 
being  a  possibly  better  hold  on  the  water,  and  a  decrease  in 
the  so-called  short-circuiting  around  the  tips  of  the  blades. 

In  regard  to  the  influence  of  any  fanciful  variation  of  pitch, 
we  have  but  to  remember  the  nature  of  the  wake  as  discussed 
in  §  41.  It  is  there  shown  that  relative  to  a  uniform  pitch 
any  propeller-blade  must  necessarily  work  with  an  excessively 
varying  distribution  of  slip,  both  over  its  surface  and  during 
a  revolution,  and  hence  relative  to  a  uniform  slip  it  must  act 
as  though  the  pitch  were  correspondingly  variable.  A  pro- 
peller of  pitch  variable  according  to  some  definite  law  will 
therefore  not  give  a  corresponding  distributed  variation  of 
slip,  and  its  actual  variation  of  pitch  is  so  small  as  to  be  en- 
tirely swallowed  up  in  the  greater  wake  variation  of  slip,  so 
that  its  whole  relation  to  the  wake  will  be  but  slightly  differ- 
ent from  that  of  a  uniform-pitch  propeller,  and  in  any  event 
entirely  different  from  that  in  a  uniform  stream. 

Any  benefit  supposed  to  arise  from  some  particular  distri- 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 

bution  of  pitch  and  slip  will  therefore  be  wholly  fanciful  or 
accidental,  since  in  the  actual  case  the  variation  of  slip  corre- 
sponding to  that  of  pitch  will  not  be  even  approximated  to. 
See  also  §§  42  and  44. 

The  Laying  Down  of  a  Screw  Propeller.  —  We  take  as 
the  simple  and  standard  case  a  propeller  of  uniform  pitch 
as  given  by  llb^cl.  We  must  have  the  following  details  given 
or  assumed  : 

Diameter. 

Pitch. 

Area. 

Diameter  and  Shape  of  Hub. 

Shape  of  Blade. 

We  may  also  assume  for  simplicity  that  the  blades  are  to 
be  cast  with  the  hub,  and  that  they  are  to  be  symmetrical 
about  a  radial  center  line. 

Let  XX,  Fig.  86,  be  the  axis  and  O  Fa  perpendicular, 
on  which  the  center  line  OB  is  taken  equal  to'  the  radius  of 
the  propeller.  The  hub  is  then  laid  off  as  in  the  figure.  The 
blade  area  being  known,  we  readily  find  the  mean  half-width, 
and  using  this  as  a  general  guide  sketch  in  a  trial  half- 
contour  AGB.  This  area  may  then  be  checked  by  planimeter 
or  otherwise,  and  adjusted  until  it  is  correct  in  amount  and 
the  contour  is  satisfactory.  The  length  of  the  root-section  is 
usually  taken  about  .9  that  of  the  maximum  width. 

The  distance  CB  is  then  divided  into  any  convenient 
number  of  parts,  measuring  from  either  B,  C,  or  O,  as  may 
be  most  convenient. 

From  (i)  the  value  of  tan  a  for  any  point  F  is 


2nr       27T  27T 


PROPELLER   DESIGN. 


315 


,  therefore,  we  take  a  point  E  such  that  OE  =  p  ~-  2nt 
the  angle  EFO  will  give  the  corresponding  value  of  a.  A 
similar  relation  holds  for  all  other  points,  so  that  if  from  E  a 

Y 


FIG.  86. 

bundle  of  lines  be  drawn  to  the  several  points  on  the  radius, 
the  inclinations  of  these  lines  to  OB  will  give  the  correspond- 
ing values  of  a. 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 

Hence  if  EH  is  taken  equal  to  FG,  it  is  evident  that  El 
is  the  projection  of  FG  on  a  longitudinal  plane,  and  HI  its 
projection  on  a  transverse  plane.  If,  therefore,  FK  is  made 
equal  to  £1,  K  will  be  the  projected  view  of  G,  and  making 
PL  =  FK,  we  have  two  points  on  the  projected  contour  of 
the  blade.  In  this  way  we  may  find  the  entire  projected 
view  of  the  blade  as  shown.  The  other  projections  are  read- 
ily found  in  a  similar  manner;  but  as  they  involve  only  an 
exercise  in  descriptive  geometry,  the  details  need  not  be  here 
explained.  It  is  readily  seen  that,  so  far  as  the  contour  is 
concerned,  this  whole  construction  is  approximate,  and  not 
accurate ;  and  it  may  be  mentioned  that  in  the  projection  on 
a  transverse  plane  the  lines  such  as  GG1  may  be  taken  as 
arcs  of  circles  with  OP  as  radius,  and  the  distances  such  as 
HI  would  then  be  laid  off  along  these  arcs,  or  practically  as 
chords. 

If  the  blade  is  not  symmetrical  about  OB,  it  may  be  laid 
off  in  any  form  desired,  and  then  the  two  parts  must  be  pro- 
jected separately,  but  otherwise  in  the  same  general  manner. 

For  detachable  blades  the  same  general  method  is  used, 
the  form  at  the  root  being  appropriately  modified  to  suit  the 
means  of  attachment  to  the  hub.  In  any  case  the  blade  joins 
the  hub,  or,  if  detachable,  its  circular  boss,  by  a  well-rounded 
fillet. 

For  blades  of  more  complicated  form  and  distribution  of 
pitch  the  same  general  principles  may  be  applied  to  find  the 
approximate  appearance  of  the  projections.  It  may  be  ob- 
served, however,  that  at  best  these  are  but  approximate,  and 
moreover  are  of  no  practical  value  to  the  pattern-maker  or 
moulder.  The  information  of  which  he  makes  direct  use  is 


PROPELLER   DESIGN. 

much  more  restricted  in  amount,  as  will  be  explained  at  a 
later  point. 

Returning  to  Fig.  86,  the  next  step  is  to  lay  down  the 
thickness  CM  at  the  root  of  the  blade.  A  straight  line,  MB, 
is  then  drawn  to  the  tip.  This  will  give  the  middle-line 
thickness  of  the  various  sections.  At  the  tip  such  line  must 
be  run  into  a  curve  as  shown,  in  order  to  insure  good  casting. 
For  bronze  propellers  such  thickness  may  be  taken  at  about  -fa 
inch  per  foot  of  diameter.  For  cast  iron  it  must  be  slightly 
thicker.  The  varying  sections  or  thickness  strips  may  now 
be  put  in.  Thus  SQ  is  made  equal  to  ST,  and  a  circular  or 
parabolic  arc  is  run  through  P,  Q,  and  R.  When  the  outer 
sections  are  reached,  as  in  Fig.  87,  such  an  arc  would  give  too 


FIG.  87. 

thin  an  edge  for  reliable  casting.  We  may,  therefore,  draw 
lines  at  the  ends  as  shown,  making  the  edge  of  the  requisite 
angle,  and  then  run  in  a  smooth  curve  tangent  to  these  lines 
at  the  ends,  and  parallel  to  AB  at  the  centre  D.  This  mini- 
mum angle  may  vary  from  as  little  as  5°  for  small  bronze 
propellers,  such  as  are  used  on  torpedo-boats  and  fast  yachts, 
to  15°  or  20°  for  cast-iron  propellers  of  large  size,  such  as  are 
used  in  ordinary  mercantile  practice.  The  smaller  angles  are 
to  be  recommended. 

The  thickness    is,  of   course,  placed   on  the  back    of  the 
blade ;  that    is,  forward  of  the  helicoidal    surface  which  has 
been  laid  out  as  the  driving-face.     For  the  sections  near  the 
root,  however,    certain    modifications   are    sometimes    made, 
:   on  the  following  considerations:   AB,  Fig.  88,  denot- 


318 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


ing  the  face,  we  have  by  the  usual  construction  EC  =  pitch 
and  DC  =  slip.  Then,  considering  the  water  as  undisturbed 
and  flowing  axially  to  the  propeller,  we  have  DO  as  the  direc- 
tion of  relative  motion.  Draw  HJ  parallel  to  DO.  Then, 
with  the  usual  form  of  section,  the  back  of  the  blade  from  B 
to  J  will  be  met  by  this  stream,  and  the  resultant  pressure 
will  have  a  component  from  forward  aft,  thus  reducing  the 


FIG.  88. 

effective  thrust.  If,  however,  the  section  is  made  as  shown 
by  the  full  line,  we  have  the  back  merely  exposed  to  frictional 
resistance,  while  the  direct  flow  meets  the  face  only,  and  may 
thus  give  a  slightly  increased  thrust.  This  change  in  section 
amounts  to  a  great  increase  of  pitch  on  the  driving-face  from 
B'  to  K.  The  simplicity  of  this  construction,  however,  is 
doubtless  far  from  representing  the  path  of  the  water  in  the 


PROPELLER   DESIGN.  319 

actual  case,  and  while  it  is  possible  that  some  such  modifica- 
tion of  the  section  at  the  root  may  be  advantageous,  we  are 
not  in  a  position  to  base  such  an  opinion  upon  experimental 
data. 

We  have  above  referred  to  the  fact  that  the  information 
actually  needed  by  the  pattern-maker  or  moulder  does  not 
include  the  various  geometrical  projections  there  referred  to. 
The  information  strictly  necessary  is  simply  the  following: 
The  principal  dimensions  and  characteristics  of  the  propeller 
and  hub ;  the  expanded  form  or  contour  of  the  blade,  as  in 
AGBG^AI,  Fig.  86;  the  shape  and  angular  position  of  the 
generatrix,  or  else  the  form  of  a  series  of  guides  at  successive 
radial  distances;  the  distribution  of  the  thickness.  The 
necessary  information  may  therefore  be  classified  under  these 
four  heads : 

(1)  The  chief  dimensions  ; 

(2)  The  contour  of  the  blade; 

(3)  The  law  of  the  pitch; 

(4)  The  distribution  of  the  thickness ; 

Thus  for  a  uniform  pitch  propeller,  the  information  given 
in  Fig.  86  aside  from  the  projection  AKB  is  all  that  is  needed 
to  prepare  a  pattern  in  wood  or  sweep  up  the  form  in  loam. 

Instead  of  the  above  method  some  designers  prefer  to  lay 
down  first  the  projection  on  a  transverse  plane,  making  the 
projected  area  a  certain  fraction  of  the  disk  area.  See,  for 
example.  Fig.  53.  The  expanded  form  and  the  other  projec- 
tion may  then  be  found  by  the  inverse  of  the  methods  given 
above. 

In  Plate  A  is  given  a  copy  of  the  working  drawing  for 
the  propellers  of  the  U.  S.  battleships  Indiana  and  Massa- 


320 


RESISTANCE  AND   PROPULSION  OF  SHIPS. 


chusetts,  showing  the  information  provided  in  these  cases  of 
naval  war-ship  design. 

Measuring  the  Pitch  of  a  Screw  Propeller. — From  (i)  we 
have/  —  2nr  tan  a.      If  therefore  at  a  given  value  of  r  the 

n 


SCREW  PROPELLER 

BATTLE 'SHIP  No  2. 


SCALE 


PLATE  A. 

value  of  tan  a  can  be  determined,  the  pitch  is  known.  Such 
measurement  requires  necessarily  the  establishment  of  a 
transverse  plane,  or  plane  perpendicular  to  the  axis  of  the 
propeller.  If  the  propeller  is  lying  on  a  floor,  the  latter  may 
be  used,  care  being  taken  that  the  propeller  is  so  blocked  up 
that  its  axis  shall  be  perpendicular  to  the  floor.  Otherwise  a 
plug  carrying  perpendicular  to  the  axis  a  swinging  arm  may 
be  fitted  to  the  shaft  hole,  thus  giving  a  movable  line  of 


PKO TELLER   DESIGN. 


321 


ference.  If  the  propeller  is  in  place  on  the  ship,  the  arm 
may  be  attached  to  the  hub  or  to  the  shaft  nut,  or  so 
arranged  in  any  way  as  to  give  a  line  of  reference  movable 
or  adjustable  in  the  transverse  plane. 

We  will  first  note  an  approximate  method  which  consists 
in  determining  at  what  value  of  r  we  have  a  =  45°,  tan 
a  =  I,  and  hence  /  =  2nr.  The  transverse  plane  being 
established  as  above,  a  bevel  or  triangle  with  angle  45°  or 
even  an  improvised  sheet  of  metal  may  be  applied,  and  the 
corresponding  location  of  r  approximately  found. 

For  more  accurate  measures  we  have  to  determine  at  a 
selected  value  of  r  two  sides  of  a  triangle,  as  in  Fig.  89.  AB 


FIG.  89. 

lies  in  a  transverse  plane  and  on  the  surface  of  a  cylinder  of 
radius  r.  AC  lies  on  the  face  of  the  blade  and  on  the  same 
cylinder,  being  in  fact  their  intersection.  BC  is  parallel 
to  the  axis.  The  triangle  ABC  developed  is  evidently 
similar  to  ADC  of  Fig.  55  for  the  same  radius  and  pitch,  the- 
relation  being  as  shown  in  Fig.  90.  If,  therefore,  we  have  an 
arm  swinging  about  the  axis,  we  may  by  running  a  line  per- 
pendicular to  this  arm  at  a  fixed  value  of  r  determine  any 
desired  number  of  points  on  the  line  AC.  The  length  of  AC 
may  then  be  measured  by  a  flexible  rule  or  batten,  or,  if  it  is 
short,  it  will  not  differ  sensibly  from  the  straight  line  joining 
its  extremities.  BC  is  then  determined  as  the  difference 


322 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


between  the  axial  distances  from  the  points  A  and  C  to  the 
transverse  plane  or  swinging  arm.     We  have  then 


BC 
sin  a  =  -r-~      and      tan  a  = 


BC 


~  BC 


and 


/=  -r& 


2nrBC 


VAC' 


FIG.  90. 

Still  otherwise,  we  may  measure  the  angle  ft  subtended  at 
the  axis  by  the  arc  AB.     We  then  have 


AB=r(S, 


and 


27TBC 

fi 


If  the  distance  AC  is  small  relative  to  the  dimensions  of 
the  propeller  it  gives  the  pitch  simply  for  that  particular 
small  part  of  the  surface,  and  if  the  pitch  is  thus  determined 
in  various  parts  of  the  surface  it  will  usually  be  found  to  vary 
very  considerably  even  in  a  propeller  intended  to  be  of 
uniform  pitch.  This  arises  from  minor  departures  from 
exactness  in  the  pattern,  mould,  and  casting.  In  rough 
work  such  variation  may  be  as  great  as  10  per  cent  on  either 


PROPELLER   DESIGN. 


323 


side  of  the  intended  value,  though  with  proper  care  the  limit 
of  variation  should  be  much  less. 

The  details  of  the  operation  of  measuring  the  pitch  will 
frequently  vary  with  the  means  at  hand  and  the  particular 
circumstances  of  the  case;  but  if  the  fundamental  geometrical 
problem  to  be  solved  is  kept  clearly  in  mind,  such  necessary 
variations  will  present  no  difficulty. 

Table  for  connecting  the  Slip  A  ngle  <f>  with  the  Slip  Ratio  s. 
— This  is  deduced  as  follows:  Referring  to  Fig.  55,  we  have 


tan  a  = 


2nr 


and 


whence,  with  the  nomenclature  there  used,  we  readily  find 


tan  0  = 


nys 


Table  I  gives  the  values  of  0  for  varying  values  of  y 
and  s. 

Table  for  Determining  the  Modification  of  Pitch  due  to  the 
Twisting  of  a  Blade. — In  propellers  with  detachable  blades  it 
is  customary  to  provide  for  twisting  the  blade  slightly  on  the 
hub,  and  thus  modifying  the  pitch.  The  amount  of  modifi- 
cation at  any  given  radial  distance  or  value  of  y,  is  found  as 
follows: 

Referring  to  Fig.  55,  we  have  in  general 

/  =  2  nr  tan  a  =  nd  tan  a, 


whence 


tan  a  =    —.-  =  - 
nd        Try 


324  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

Let  ft  be  the  amount  of  twist.     Then,  evidently, 


and  A 


/0  tan  a 

Since  ft  is  always  quite  small,  this  may  be  put  in  the  form 

/,  _  i  ±  ft  cot  OL        i  ±  /?TTJ/ 
A  ~  I  =F  A  tan  a  "  ^_  ' 

*r 

Table  II  gives  values  of/!  -r-/0  for  varying  values  of  /? 
and  jj/. 

It  is  thus  seen  that  the  amount  of  change  varies  consider- 
ably for  different  portions  of  the  blade.  Hence  if  the  pitch 
were  uniform  before  twisting,  it  cannot  be  so  afterward. 
The  twist  will  have,  in  fact,  transformed  the  surface  from  one 
of  uniform  pitch,  to  one  with  a  more  or  less  irregular  distri- 
bution from  root  to  tip  of  blade.  This,  of  course,  is  by  no 
means  necessarily  prejudicial,  and  so  long  as  the  angle  of  twist 
is  small  we  have  no  reason  experimentally  to  expect  any 
particular  effect  either  good  or  bad  due  merely  to  the  irregu- 
larity of  the  pitch.  The  twisting  thus  provided  for  is  for  the 
purpose  of  changing  the  effective  or  mean  pitch  of  the  pro- 
peller as  a  whole,  and  the  above  considerations  indicate  that 
so  long  as  the  amount  of  twist  is  small,  we  have  no  experi- 
mental basis  for  objecting  to  this  mode  of  obtaining  such 
modification. 

As  a  final  question  we  may  ask  what  will  be  the  mean 
pitch  of  the  blade  thus  modified.  This  will  depend  on  the 
definition  of  mean  pitch  as  discussed  in  §  42,  to  which  refer- 
ence may  be  made. 


PROPELLER   DESIGN. 


325 


TABLE  I. 

VALUE   OF  THE   SLIP  ANGLE   0. 


Slip. 

.05 

.10 

•  15 

.20 

•  25 

•3° 

'35 

.40 

Diameter 

Ratio. 

.1 

o  51 

I   48 

2   51 

4  oo 

5  17 

6  44 

8   21 

IO   12 

.2 

21 

2  47 

4  20 

6  oo 

7  49 

9  46 

ii  53 

14   II 

-3 

28 

3  oi 

4  39 

6   22 

8  ii 

10  06 

12   06 

14   13 

•4 

25 

2  54 

4  26 

6  02 

7  41 

9  24 

II   10 

12  59 

-5 

19 

2   40 

4  04 

5  30 

6  58 

8  28 

10   00 

ii  35 

.6 

12 

2   25 

3  41 

4  57 

6  15 

7  34 

8  56 

10  18 

•7 

5 

2   12 

3  19 

4  28 

5  38 

6  48 

7  59 

9  12 

.8 

00 

2   00 

3  oi 

4  03 

5  05 

6  08 

7  12 

8  16 

•9 

o  54 

I  49 

2  45 

3  4i 

4  37 

5  34 

6  32 

7  30 

I.O 

o  50 

i  40 

2  31 

3  22 

4  14 

5  06 

5  5» 

6  51 

TABLE  II.* 

FACTOR  FOR  MODIFICATION  OF  PITCH  DUE  TO  TWISTING  OF 

BLADE. 


Twist 

of 

i° 

2° 

3° 

4° 

5° 

6° 

Blade. 

Factor  for 

Factor  for 

Factor  for 

Factor  for 

Factor  for 

Factor  for 

8.2 

B  ^8 

1 

i 

§ 

j 

i 

i 

rease. 

ease. 

rease. 

ease. 

rease. 

«J 

• 

rt 

I 

£X 
Q 

o 
t: 

<L> 

Q 

I 

£ 

| 

K 

Q 

0 

^c 

I 

Q 

c 

u 

a 

u 
o 

c 

u 

Q 

.1 

-065 

•943 

.138 

.891 

.221 

.843 

1-315 

.800 

1.425 

.76i 

•553 

•725 

.2 

.040 

•9&3 

.083 

•927 

.127 

•893 

1-175 

.861 

1.226 

.830 

.281 

.800 

•3 

.036 

.966 

•073 

•933 

.III 

.QOI 

.152 

.870 

.194 

.840 

•237 

.811 

•4 

•037 

•9^5 

.074 

•930 

.  112 

.897 

.152 

.864 

•193 

.832 

.236 

.801 

•5 

•039 

</>2 

.079 

•925 

•  I2O 

.886 

.162 

.852 

•  205 

.817 

.249 

•783 

.6 

•043 

-958 

.o8b 

.918 

.130 

.877 

.176 

•837 

.222 

.798 

.269 

.760 

•7 

•047 

•954 

.094 

.909 

.142 

.864 

.192 

.820 

.242 

•777 

•293 

•734 

.8 

•051 

.950 

.103 

.900 

.156 

.851 

.210 

.802 

.264 

•754 

•  320 

.707 

•9 

.o=,6 

•945 

•"3 

.891 

.170 

•837 

.229 

.783 

.288 

.730 

•348 

.678 

I.O 

.060 

.940 

.122 

.880 

I84 

.821 

.247 

.763 

•3" 

•705 

•376 

.648 

Taylor's  "  Resistance  of  Ships  and  Screw-propulsion."  Table  XV. 


CHAPTER   V. 
POWERING    SHIPS. 

56.  INTRODUCTORY. 

THE  various  constituents  of  the  power  which  it  is  neces- 
sary to  develop  in  the  cylinders  of  a  marine  engine  in  order  to 
propel  a  ship  at  a  given  speed  have  been  already  discussed  in 
§  46.  We  have  now  to  consider  the  various  suppositions 
which  will  enable  us  to  make  an  estimate  of  the  total  power 
thus  required  in  any  given  case.  These  are  of  two  general 
classes  and  lead  to  two  general  methods  for  the  estimate  of 
power. 

(i)  An  assumption  of  the  necessary  constants  which  will 
enable  us  to  compute  the  resistance  and  the  power  required 
to  overcome  it,  together  with  that  required  for  the  various 
other  constituents  going  to  make  up  the  I.H.P.  This  also 
may  be  done  in  either  of  two  ways: 

(a)  We  may  make  the  various  assumptions  necessary  to 
compute  the  different  constituents  individually.  This  in- 
volves the  following  elements:  the  power  for  the  resistance 
proper  or  the  E.H.P. ;  the  amount  involved  in  the  augmen- 
tation of  resistance;  the  wake  factor;  the  propeller  efficiency; 
and  the  amount  absorbed  by  friction.  By  a  proper  combina- 
tion of  these  as  indicated  by  Fig.  66  we  readily  work  back  to 

the  total  I.H.P. 

326 


POWERING   SHIPS. 


327 


(&)  We  may  make  the  assumptions  necessary  to  compute 
the  E.H.P.  as  above,  and  then  assume  the  propulsive 
coefficient  or  ratio  E.H.P.  -r-  I.H.P.,  whence  the  latter  fol- 
lows directly  from  the  former,  and  without  special  assump- 
tions as  to  its  other  individual  elements. 

(2)  The  second  general  method  involves  certain  assump- 
tions which  justify  the  extension  of  the  law  of  comparison 
from  resistance  to  power,  and  thus  enable  us  to  compute  the 
I.H.P.  in  any  given  case  from  that  actually  observed  for 
similar  ships  at  corresponding  speeds. 

We  shall  discuss  these  methods  in  order. 


57.  THE  COMPUTATION  OF  WP,  OR  THE  WORK  ABSORBED 
BY  THE  PROPELLER. 

The  necessary  information  and  assumptions  requisite  for 
the  computation  of  R  have  been  discussed  in  Chapter  I  and 
in  §  45.  The  amount  of  augmentation  of  resistance  must  be 
taken  by  judgment  guided  by  such  information  relating  to 
similar  cases  as  may  be  available.  Using  the  E.H.P.  as  a 
base,  the  additional  amount  of  power  thus  required  will 
usually  be  found  between  10  and  25  per  cent.  Definite 
information  with  regard  to  the  amount  of  this  constituent  of 
I.H.P.  is  only  generally  obtainable  by  model  experiments  as 
referred  to  in  §  30.  With  such  appliances,  the  true  and 
augmented  resistances  are  readily  found. 

Unless  there  is  available  some  definite  information  based 
on  model  experiments  relative  to  the  probable  amount  of 
augmentation,  there  is  little  use  in  attempting  its  independent 
estimate. 

The  determination  of  the  wake  factor  w  has  been  already 


328  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

referred  to  in  §§  44  and  49.  With  the  wake  factor  and 
augmentation  known  or  assumed,  we  readily  pass,  as  ex- 
plained in  §  46,  from  CF  to  Cf,  Fig.  66. 

If,  instead  of  estimating  augmentation  and  wake  factor 
separately  we  should  consider  the  ship  efficiency  (§  46)  as  i, 
then  we  have,  in  Fig.  66,  CF  =  C^E  and  propeller  power  or 
BE  =  CF  divided  by  propeller  efficiency. 

The  considerations  relating  to  propeller  efficiency  have 
been  discussed  in  §§  49  to  51.  Guided  by  such  information 
as  is  available  and  as  seems  applicable  to  the  case  in  hand,  its 
value  is  assumed.  This,  as  we  have  seen,  may  in  usual  cases 
be  expected  to  lie  between  65  and  69  per  cent. 

58.  ENGINE  FRICTION. 

We  come  next  to  the  amount  of  power  absorbed  by  fric- 
tion between  the  propeller  and  the  cylinders. 

We  may  remark  at  the  beginning  that  there  is  very  little 
exact  or  satisfactory  information  on  these  points.  Model 
experiments  are  here  of  no  avail,  at  least  for  purposes  of 
actual  measurement.  Such  measurement  would  require  at 
the  various  speeds  the  simultaneous  determination  of  the 
power  in  the  cylinders  and  of  the  power  received  at  the  pro- 
peller. If,  for  example,  just  forward  of  the  propeller  there 
could  be  inserted  a  transmission  dynamometer  which  would 
measure  the  turning  moment,  then  the  product  of  this  by  2n 
times  the  number  of  revolutions  per  minute  would  give  the 
power  transmitted  to  the  propeller.  Such  determinations 
are  obviously  difficult  and  costly,  and  hence  more  or  less 
indirect  methods  are  resorted  to. 

In    the   first   place   it   is    assumed   that   the    total   power 


POWERING    SHIPS. 


329 


absorbed  by  friction  may  be  expressed  by  an  equation  of  the 


form 


(i) 


where  h  is  a  constant  for  any  given  engine,  N  is  the  number 
of  revolutions,  and  h  N  is  the  amount  due  to  what  is  termed 
the  initial  friction,  while  W  is  the  I.H.P.  and  /is  likewise  a 
constant  for  the  given  engine. 

The  supposition  that  the  total  frictional  power  may  be 
thus  expressed  is  quite  arbitrary,  but  seems  to  answer  the 
purpose  required  as  closely  as  present  data  can  determine. 
It  assumes  simply  that  there  will  exist  at  the  various  joints 
and  rubbing-surfaces  a  certain  tangential  resistance  of  which 
a  part  proportional  to  h  is  sensibly  independent  of  revolu- 
tions or  load,  and  represents  what  is  termed  the  initial  fric- 
tion. The  existence  of  this  factor  is  chiefly  due  to  the 
pressure  at  the  various  stuffing-boxes,  piston-rings,  etc.,  and 
to  the  weight  of  the  shafting  and  other  members  supported 
in  bearings.  This  part  of  the  total  friction  is  also  sometimes 
known  as  that  due  to  the  dead  load.  The  work  necessary  to 
overcome  such  a  constant  resistance  will  vary  directly  as  the 
revolutions,  and  hence  will  be  represented  by  a  term  of  the 
form  hN.  It  is  further  assumed  that  the  remainder  of  the 
tangential  resistance  at  the  rubbing-surfaces  will  vary  directly 
as  the  load  on  the  engine,  or  as  the  mean  effective  pressure, 
and  hence  the  work  necessary  to  overcome  this  part  will  be 
proportional  to  the  product  of  mean  effective  pressure  and 
revolutions,  or  to  total  power  developed.  Hence  this  part 
of  the  frictional  work  will  be  represented  by  a  term  of  the 
form  IW. 

Both  of  these  terms  may  be  expected  to  vary  with  the  type 
and  condition  of  the  engine,  and  with  the  presence  or  absence 


33°  RESISTANCE  AND    PROPULSION   OF  SHIPS. 

of  attached  pumps.  In  general  the  friction  of  horizontal  en- 
gines is  greater  than  that  of  vertical.  The  power  absorbed 
in  friction  also  increases  in  general  with  the  number  of  cylin- 
ders and  with  the  multiplication  of  rubbing-joints.  The  fric- 
tion of  new  engines  is  also  naturally  greater  than  when  they 
have  become  somewhat  worn.  Again,  incorrect  adjustment 
or  alignment  either  at  the  beginning  or  due  to  wear  or  work- 
ing of  the  ship  may  give  rise  to  excessive  frictional  loss.  In 
such  cases  the  condition  is  likely  to  manifest  itself  by  hot 
bearings  and  other  signs  that  something  is  wrong  at  the  points 
thus  affected.  The  presence  of  attached  air-pumps  occasions 
an  absorption  of  power  not  only  to  overcome  the  friction  of 
the  pump,  but  for  its  regular  running,  which  amount  must  be 
subtracted  from  the  I.H.P.  in  order  to  properly  obtain  that 
sent  to  the  propeller.  This  item  is  not  to  be  viewed  as  a 
loss,  but  simply  in  such  cases  as  a  constituent  of  the  total 
power  which  must  be  eliminated  in  order  to  determine  the 
net  amount  delivered  to  the  propeller.  With  independent 
pumps  this  item,  of  course,  does  not  exist  as  a  constituent  of 
the  power  of  the  main  engines. 

The  power  required  to  operate  pumps  has  been  deter- 
mined by  various  special  and  individual  trials.  The  power 
absorbed  by  the  initial  friction  has  been  determined  by  slowly 
running  the  unloaded  main  engine  by  the  turning  engine, 
or  by  derivation  from  curves  of  revolution  and  power  as 
described  under  the  next  sub-head.  The  power  absorbed  by 
the  load  friction  has  been  estimated  from  various  coefficients 
derived  from  stationary  engines  in  which  the  different  quan- 
tities are  susceptible  of  measurement,  and  from  analyses  of 
the  total  I.H.P.  in  which  the  various  efficiencies  and  ratios 
are  derived  by  comparison  from  model  experiments. 


POWERING   SHIPS. 


33' 


Determination  of  Initial  Friction  from  Speed-trial  Data. — 
In  §  47  we  have  defined  the  reduced  mean  effective  pressure, 
and  we  may  evidently  consider  this  as  balanced  against  the 
mean  total  resistance  to  the  movement  of  the  engine.  This 
total  resistance  may  be  considered  as  the  sum  of  two  parts, 
one  due  to  the  useful  load  and  the  other  due  to  the  friction 
load. 

We  have  already  referred  to  the  frictional  or  tangential 
resistance  at  the  joints  and  rubbing-surfaces  under  two  heads: 
that  due  to  initial  conditions  and  that  due  to  the  load,  corre- 
sponding to  initial  and  load  friction.  Evidently  the  frictional 
work  will  require  an  additional  pressure  on  the  pistons  or  an 
additional  amount  of  reduced  mean  effective  pressure  as 
above  defined,  and  such  additional  amount  may  naturally  be 
considered  in  two  parts,  one  due  to  initial  friction  and  the 
other  to  the  load  friction.  We  may  thus  consider  the  entire 
reduced  mean  effective  pressure  as  made  up  of  three  parts, 
one  corresponding  to  the  net  or  delivered  power  and  two  due 
to  friction  as  above.  Denoting  these  respectively  by  /, /,, 
/,,  we  have 


/=A+A+A 


(2) 


From  the  nature  of  initial  friction  as  above  defined  it 
follows  that/,  is  constant  and  is  in  fact  proportional  to  the 
factor  //  there  used,  while  /,  and  also  p  will  vary  with  the 
load. 

If  now  we  have  a  series  of  corresponding  values  of  H  and 
N,  we  may  derive  by  §  47  (3)  the  values  of/.  Plotting  these 
on  N  as  an  abscissa,  we  shall  have  a  curve  as  in  Fig.  91.  Now 
when  TV—  o,  /,  and  /  must  necessarily  be  o,  and  hence/  will 
be  equal  to/,.  If  therefore  the  curve  be  extended  back  by 


332 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


judgment,  it  will  cut  the  axis  of  Y  at  a  point  A  which  will 
give  an  approximation  OA  to  the  value  of/,.     Then  for  the 


t>  VALUES  OF  N  * 

FIG.  91. 

general  value  of  the  initial  friction  power  at  any  number  of 
revolutions  N  we  have 


==  hN,  as  above. 


33000 


Hence  we  see  that 


33000 


In  order  that  such  an  approximation  may  be  of  value, 
special  care  should  be  taken  with  the  determinations  relating 
to  the  lower  points  on  the  curve,  and  the  revolutions  should 
be  reduced  to  the  lowest  number  at  which  the  engine  can  be 
kept  in  continuous  movement. 

We  may  also  for  the  same  result  use  the  same  data  some- 
what differently  as  follows: 


POWERING    SHIPS. 


333 


From  the  common  horse-power  formula  we  have 

_  2pLA 


33000 


Hence 


33000 


Hence  if  H  be  plotted  on  TV  as  an  abscissa  we  shall  have  a 
curve  as  in  Fig.  92.      If  this  be  extended  back  through  O  by 


VALUES  OF  N 
FIG.  92. 


judgment,  it  will  have  at  O  a  certain  inclination  to  OX,  and 
the  tangent  of  this  angle   or  of  BOX  will   therefore  be   an 


approximation  to  the  value  of  ~^-     ;  and  from  this  the  corre- 

Jo 


spending  value  of/,  may  be  found.     See  also  §  78  for  another 
method  of  finding  the  initial  friction. 

The  existing  data  relating  to  initial  friction  gathered  in 
this  and  other  ways  indicate  that  in  amount  it  may  be 
expected  to  lie  between  5  and  8  per  cent  of  the  full  power  of 


334  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

the  engine,  or  that/,  will  be  from  5  to  8  percent  of  the/  for 
full  load.  This  would  correspond  to  a  value  of  from  2  to  3 
pounds  per  square  inch  on  the  low-pressure  piston  for  triple- 
expansion  engines  under  usual  conditions  as  above  noted. 
Let  this  percentage  ratio  be  denoted  by  i  and  denote  full- 
power  conditions  by  accents.  Then  we  should  have 
Work  of  initial  friction  at  full  load  —  hN'  =  iH', 

.Hf 
or  k=  i^-r ; 

and  work  for  initial  friction  in  general  =  hN '=.  ,  where, 

as  above  stated,  we  may  expect  i  to  be  found  between  .05 
and  .08.  These  figures  relate  more  especially  to  the  main 
engines  alone. 

Value  of  the  Load  Friction. — We  have  above  stated  that 
this  is  taken  as  proportional  to  the  whole  power  H.  It  might 
seem  more  correct  to  take  it  as  proportional  to  the  total 
power  minus  the  initial  friction.  The  difference  involved  is, 
however,  quite  negligible  in  view  of  the  general  uncertainty 
surrounding  the  whole  question,  and  we  naturally  prefer, 
as  the  simpler  of  the  two,  the  method  as  stated.  We  have 
therefore 

Load  friction  =  IH. 

The  existing  data  relating  to  load  friction  indicate  that  / 
may  be  expected  to  vary  between  nearly  the  same  limits  as  i, 
or  from  .05  to  .08. 

General  Equation  for  Friction. — Combining  the  two  parts., 
we  have 

Total  friction  =  -—  +  Iff. 


POWERING    SHIPS, 


335 


If  we  plot  the  total  power  and  the  power  absorbed  by 
these  two  components  of  friction  on  revolutions  as  an  abscissa, 
we  shall  have  a  diagram  as  in  Fig.  93,  where  the  ordinates 


VALUES  OF  N 

FIG.  93. 

itween  X  and  OA  give  the  values  of  the  initial  friction, 
those  between  OA  and  OB  the  values  of  the  load  friction,  and 
those  between  OB  and  OC  the  values  of  the  net  or  delivered 
power.  This  equation  and  diagram  serve  to  illustrate  the 
increasing  relative  importance  of  friction  at  reduced  speeds. 
Thus  suppose  at  full  power  the  friction  is  taken  at  14  per 
cent,  equally  divided  between  load  and  initial  friction. 
Denoting  full  power  by  100,  we  should  then  have  for  one- 
half  speed  approximately  one-eighth  power  or  12.5.  The 
load  friction  would  be  reduced  in  the  same  proportion,  and 
would  therefore  become  7  -i-  8  =  .875.  The  initial  friction 
would  be  reduced  in  the  ratio  of  the  revolutions  or  approxi- 


336  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

mately  to  one  half  its  full-power  value,  or  to  3.5.  Hence  the 
total  friction  will  be  4.375,  and  the  percentage  4.375  -=-  12.5 
=  35  per  cent.  This  illustrates  forcibly  the  wastefulness  of 
running  at  reduced  power,  and  the  impossibility  of  obtaining 
good  propulsive  efficiencies  under  these  conditions. 

At  full  power,  therefore,  we  may  expect  that  the  friction 
of  a  marine  engine  will  absorb  an  amount  lying  not  far  from, 
say,  15  per  cent  as  a  mean  value.  With  the  best  modern 
design  and  construction  this  figure  may  fall  to  12  per  cent  or 
even  possibly  to  10  per  cent,  while  with  older  engines  or 
poorer  construction  or  lack  of  adjustment  it  may  rise  to  18  or 
20  per  cent  or  even  more. 

It  should  be  understood  that  the  division  of  the  power 
absorbed  by  the  friction  as  indicated  in  the  present  section  is 
not  altogether  satisfactory,  and  the  only  excuse  for  its  use  is 
our  ignorance  of  a  more  correct  analysis.  It  is  not  to  be 
expected  that/a  will  be  absolutely  constant  at  all  revolutions 
or  that  /„  will  vary  exactly  with  the  load.  Further  experi- 
mental investigation  is  much  needed  on  these  points. 

59.  POWER  REQUIRED  FOR  AUXILIARIES. 

In  the  present  work  we  are  fundamentally  concerned,  of 
course,  with  the  power  needed  for  propulsion.  Inasmuch, 
however,  as  we  not  infrequently  meet  with  air-pumps  attached 
to  the  main  engine,  as  referred  to  in  the  last  section,  and  also 
with  estimates  of  power  or  power  data,  including  that  required 
for  auxiliaries,  it  will  be  well  to  note  briefly  the  power  which 
the  usual  auxiliaries  may  be  expected  to  absorb. 

There  has  been  a  great  reduction  in  this  amount  within 
the  past  few  years.  Auxiliary  machinery  has  been  so  im- 


POWERING   SHIPS. 


337 


proved  in  design  and  construction  that  its  cost  in  power 
required  for  operation  is  much  less  than  it  was  from  ten  to 
twenty  years  ago.  At  the  earlier  period  from  5  to  8  per  cent 
of  the  I.H.P.  was  believed  to  be  absorbed  by  the  resistance 
of  the  various  pumps,  most  of  which  were  attached  to  the 
main  engine.  Blechynden,*  writing  for  a  period  about  ten 
years  since,  gives  the  following  estimate,  based  on  data  derived 
mostly  from  merchant  steamers  with  triple-expansion  engines: 


Dead  load  and  air-pump. . .    7.8 

Circulating-pump 1.5 

Feed-pump 6 

Bilge-pump $ 


Per  cent  of  I.H.P. 


If  we  take  an  estimate  for  the  air-pump  of  about   I   per 
cent,  which  would  not  be  far  in  error  for  the  cases  involved, 

Ie  should  have  3.6  per  cent  required  for  all  auxiliaries, 
hese  figures  agree  generally  with  those  obtained  in  this 
>untry  about  ten  years  ago,  especially  in  the  early  vessels  of 
the  modern  steel  navy.  Taking  more  recently  the  results  of 
the  latest  practice,  especially  in  war-ship  design,  we  have  the 
following  results: 


Air-pump  . : I  to     .2 

Circulating-pump 2  to     .5 

Feed-pump 5  to     .6 

Blowers 5  to  1 . 5 


>-  Per  cent  of  I.H.P. 


Omitting  the  item  for  blowers,  we  have  for  the  remaining 
three  items  from  .8  to   1.3  per  cent,  or  slightly  over   I   per 


*  Transactions  N.  E.  Coast  Inst.  Engineers  and  Shipbuilders,  vol.  vn. 
p.  194. 


33$  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

cent  as  a  mean  value.  The  chief  improvement  has  been  in 
air-  and  circulating-pumps,  which  together  a  few  years  ago 
absorbed  from  2  to  3  per  cent,  while  with  the  latest  practice 
this  figure  has  been  reduced  to  about  one  quarter  of  this 
amount.  The  item  for  bilge-pumps  in  earlier  estimates  is  not 
so  often  met  with  now,  having  decreased  to  almost  a  negli- 
gible amount — probably  less  than  .  I  per  cent.  The  amount 
absorbed  by  blowers  is  naturally  variable  between  wide 
limits,  being  dependent  on  the  degree  of  forced  draught  and 
other  circumstances. 

It  may  be  doubted  whether  the  power  of  an  attached  air- 
'pump  can  be  brought  to  as  small  a  figure  as  for  an  independ- 
ent pump.  This  is  due  chiefly  to  the  fact  that  the  attached 
pump  usually  runs  faster  than  the  independent,  and  faster 
than  is  needed  to  maintain  a  vacuum.  In  consequence  the 
power  is  necessarily  greater,  and  for  such  cases  probably  not 
less  than  .5  per  cent  will  be  required. 

60.  ILLUSTRATIVE  EXAMPLE. 

We  will  now  illustrate  the  application  of  the  preceding 
sections  to  the  computation  of  the  I.H.P.  in  a  given  case, 
assuming  where  necessary  the  various  values  involved. 

Referring  to  Fig.  66,  let  true  resistance  =  R. 

Then  E.H.P.  =  CF  =  Ru. 

Let  the  factor  of  augmentation  be  1.16. 

Then  actual  resistance  =  I.  \6R. 

And  thrust  horse-power  =  CE  =  i.i6Ru. 

Let  true  propeller  efficiency  =  .67. 

Let  wake  factor  w  =  .15. 

Then  (i  +  w)  =  1.15. 


POWERING   SHIPS.  339 


And  horse-power  C^E  =  i.\6Ru  ~  1.15  = 
Then    propeller    horse-  power  =  BE  =   i.oiRu  -=-  .67  = 
i.  5  i  Ru. 

Next,  for  the  frictional  loss  let  i  and  /  each  =  .07. 
Then  I.H.P.  =  AE  =  \.$iRu  -f-  .86  =  1.756^. 


. 


Propulsive  coefficient  :  =  - 


l 


tese  suppositions  correspond  to  the  following  subdivision 
of  the  total  I.H.P.  in  percentages. 

Initial  friction =  .07 

Load  friction =  .07 

1.16 

Thrust  horse-power  =  -^ =  .661 

Propeller  loss  decreased  by  wake  gain.  —  .199 


Total —  i  .000 

in  §  56  (i)  (b),  the  propulsive  coefficient  were  to  be 
assumed  directly,  we  should  in  the  preceding  example  divide 
Ru  by  .57,  thus  giving  \."jtf>Ru  at  once. 

Unless  special  data  are  available  for  the  estimate  of  the 
coefficients  and  ratios  involved  in  the  process  above,  it  will 
be  found  usually  quite  as  satisfactory  to  assume  the  propul- 
sive coefficient  directly.  This  again  can  only  be  done  intelli- 
gently when  experimental  data  are  at  hand  from  somewhat 
similar  ships  under  generally  like  conditions.  The  range  of 
values,  it  will  be  remembered,  is  usually  from  .50  to  .60. 


340        RESISTANCE  AND  PROPULSION  OF  SHIPS. 

61.  POWERING  BY  THE  LAW  OF  COMPARISON. 
The  assumptions  involved  are  as  follows: 

(1)  It  is  assumed  that  the  entire  resistance  is  subject  to 
the  law  of  comparison.     The  propriety  of  this  assumption  has 
been  considered  in  §  26,  to  which  reference  may  be  made. 
As  there  shown,  the  error  is  quite  small,  and  may  be  consid- 
ered as  well  within  the  limit  of  general  uncertainty  surround- 
ing the  entire  problem.      Further,  with  a  slightly  roughened 
bottom,  as  actually  exists  for  most  of  the  time,  the  index  of 
the  speed  in  the  term  for  skin-resistance  will  approach  the 
value  2.00,  and  the  error  will  correspondingly  decrease. 

(2)  It  is  assumed  that  the  term  similar  is  extended  to 
include  ship,  propeller,   and    machinery.     This   involves    the 
general  geometrical  similarity  between  the  ships  and  between 
the  propellers,  the  ratio  being,  of  course,  the  same  for  each. 
Geometrical  similarity  is  not  necessary  for  the  machinery,  but 
it  should  be  of  the  same  general  type  in  each  case,  or  at 
least   there   should   be   good   ground  for  assuming  an   equal 
engine  friction  in  each  case. 

(3)  It  is    assumed   that  the    term  corresponding  speeds  is 
extended   to   cover   the   speeds  of   the  propellers  relative  to 
their  ships  and  to  each  other,  as  well  as  the  speeds  of  the 
ships  relative  to  the  water. 

Hence,  as  in  §  53,  we  shall  have  similar  points  on  the  two 
propellers  describing  similar  paths  with  velocities  in  the  ratio 
/\>,  and  as  there  shown  we  shall  have  also 


^     (A  V/2_  _ 

N,"  \LJ      ~  A,1/* 


(4)  It    is   assumed    if   (2)   and   (3)    are   fulfilled    that    the 
thrusts  will  bear  the  same  relation  to  the  resistance  in  each 


POWERING   SHIPS.  341 

case,  that  the  slips  true  and  apparent  will  be  the  same  per  cent 
in  the  two  cases,  that  the  efficiencies  will  be  the  same,  and 
likewise  the  augmentation  and  wake  factors.  It  is  assumed, 
in  short,  that  the  entire  relation  of  the  propeller  to  the  ship 
will  be  the  same  in  the  two  cases. 

The  propriety  of  these  assumptions  evidently  depends  on 
the  same  general  basis  as  the  application  of  the  law  of  com- 
parison to  the  resistance  of  the  ship  itself. 

As  a  result  of  the  preceding  assumptions  it  follows  that 
similar  ships  with  similar  propellers  at  corresponding  speeds 
will  have  equal  propulsive  coefficients. 

Or  otherwise  we  may  less  safely  consider  that  indepen- 
dent of  the  fulfillment  of  (2)  and  (3),  the  propulsive  efficien- 
cies and  relations  are  the  same  in  the  two  cases. 

Let  the  subscripts  I  and  2  denote  the  two  cases.  Then 
we  have 

R^  and  R^  are  the  two  resistances; 

n  a  and  ul  are  the  two  speeds; 

/2  and  /,  are  any  two  similar  dimensions; 

A  =  /a  -f-  /j  is  the  linear  ratio  ; 

a  =  the  propulsive  coefficient; 

//,  and  H  are  the  two  powers. 

Then 


and 


Now  we  have  in  §  26  a  series  of  expressions  for  7?,  -f-  Rt 
in  terms  of  various  functions  of  the  ship  and  of  the  speeds. 
Substituting  these,  and  remembering  that  «,  ~  ut  =  y\  = 


342  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

4//a  -T-  A,  vve  readily  find  the  following  series  of  expressions 
for  HI  -i-  //,,  the  ratio  between  the  powers  necessary  to  propel 
similar  ships  at  corresponding  speeds: 


_  f.v  _  M.V  _  (vx_  /£.y_  /  A}  K  V_(AW/,V 

T~\#        W  :=\J7  :"W       \Dj\lJ~-\Dj\l) 


\ij\uj  "w  V 

It  is  obvious  that  the  list  of  expressions  in  which  the  law 
of  comparison  may  be  thus  represented  is  by  no  means 
exhausted,  and  that  all  are  equivalent  and  equally  correct,  and 
will  give  identical  results,  provided  the  fundamental  condi- 
tions are  fulfilled;  i.e.,  if  the  ships  are  similar  and  the  speeds 
corresponding. 

If  therefore  we  know  the  data  in  any  one  case  it  is  a  sim- 
ple matter  to  derive  the  value  for  a  similar  ship  at  a  corre- 
sponding speed. 

As  an  illustration,  let 

Dl  =  2000; 
H,  =  3000 ; 
ul  —  16; 
Dz  —  3200. 

In  §  26  various  expressions  have  been  given  for  corre- 
sponding speeds.  Here,  having  given  only  Dl  and  D^  we  take 


naturally  ut  =  «i~         =  1.085*,  =  I7-4- 


Then     H>  =  H=  3000  X  I      •   X 


, 

In  a  similar  way,  taking  other  expressions  for  ~,  the  same 


POWERING   SHIPS. 

value  would  be  found.  Hence  the  similar  ship  of  displace- 
ment 3200  at  17.4  knots  would  require  5230  I.H.P. 

Assuming  that  the  similarity  between  the  two  ships  is 
perfect,  the  two  chief  sources  of  error  in  the  use  of  this 
method  are  in  the  extension  of  the  law  of  comparison  to  skin- 
resistance,  and  in  the  possibility  of  a  difference  in  the  propul- 
sive coefficients  in  the  two  cases.  The  nature  of  the  error 
due  to  the  first  of  these  has  been  discussed  above  and  in  §  26. 
In  regard  to  the  latter  it  may  be  seen  that  if  the  speed  of  the 
first  ship,  for  example,  is  low  or  far  from  the  normal  speed 
the  propulsive  coefficient  will  probably  be  poor,  and  less  than 
might  be  properly  expected  for  the  second  ship  if  near  her 
normal  speed  The  use  of  the  power  of  the  first  ship  in  such 
a  case  would  therefore,  other  things  being  equal,  result  in  an 
overestimate  of  the  power  for  the  second. 

In  all  such  cases,  therefore,  judgment  must  be  used  as  to 
what  extent  the  necessary  conditions  are  fulfilled,  and  as  to 
the  amount  of  reliance  to  be  placed  on  the  values  resulting 
from  any  given  comparison. 

62.  THE  ADMIRALTY  DISPLACEMENT  COEFFICIENT. 

Among  the  various  formulae  for  powering  which  have  been 
in  use  for  many  years,  none  has  obtained  so  wide  and  general 
acceptance  as  that  involving  the  use  of  the  so-called 
Admiralty  displacement  constant  or  coefficient.  This  formula 
dates  from  a  period  long  anterior  to  the  introduction  of  the 
law  of  comparison,  and  we  shall  find  it  of  interest  to  examine 
it  in  the  light  of  that  law. 

The  formula  is  expressed  by  the  equation 

ZJV 


344 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


In  practice  it  was  found  that  there  was  a  general  tendency 
to  constancy  in  the  values  of  K,  ranging  as  they  usually  did 
from  200  to  250.  The  use  of  the  formula  required  simply 
the  estimate  of  the  constant  suited  to  the  ship  and  to  the 
speed.  This,  of  course,  was  a  matter  of  considerable  uncer- 
tainty except  as  experience  in  similar  cases  furnished  data  as 
a  basis.  The  values  of  K  naturally  vary  for  the  same  ship  at 
different  speeds,  or  for  different  ships  at  the  same  speed. 
These  values  are  also  seen  to  vary  directly  as  the  efficiency  of 
propulsion.  That  is,  in  any  given  case  the  greater  the  value 
of  K  the  less  the  value  of  //,  and  hence  the  greater  the  pro- 
pulsive efficiency.  In  numerous  cases  in  which  progressive 
speed  trials  have  been  made  the  results  for  the  value  of  K 
are  similar  to  those  shown  in  Fig.  94.  At  low  speeds  the 


£300 


£100 


A  

—  • 





—i                "— 



^V 

B 

10          12          U          16          18          20 
SPEED 

FIG.  94. 


value  is  low,  due  to  the  general  mechanical  inefficiency  of  both 
engine  and  propeller  at  powers  and  speeds  far  below  those  for 
which  they  were  designed.  As  the  speed  increases  the  value 
of  K  increases,  indicating  that  due  to  an  increasing  efficiency 
the  total  power  is  related  to  the  speed  by  an  index  of  u  less 
than  3.  This  continues  to  some  speed  at  which,  on  this 
basis,  the  propeller  is  the  most  efficient  and  K  has  its  maxi- 
mum value.  Then  as  the  speed  is  increased  the  resistance 
begins  to  increase  excessively,  and  perhaps  the  propulsive 


POWERING   SHIPS. 


345 


etticiency  begins  to  decrease.  The  result  is  that  H  becomes 
related  to  //  with  an  index  greater  than  3,  and  we  have  a 
corresponding  fall  in  the  value  of  K. 

So  long  as  speeds  were  moderate  and  abundant  informa- 
tion was  available  relating  to  nearly  equal  ships  at  nearly  the 
same  speeds,  the  use  of  this  coefficient  was  reasonably  satis- 
factory. Its  assumption,  however,  was  almost  entirely  a 
matter  of  judgment,  and  there  were  few  guiding  principles  by 
means  of  which  any  definite  value  might  be  determined  as 
suitable  in  any  given  case. 

The  use  of  the  formula  in  this  way  has  therefore  in  recent 
years  fallen  into  disfavor,  and  it  is  well  understood  that  thus 
employed  it  can  only  be  expected  to  give  a  rough  approxi- 
mation to  the  power  suitable  in  any  proposed  case. 

Let  us  now  examine  the  relation  'of  this  formula  to  the 
law  of  comparison.  We  have  in  one  case 


and  in  another: 


Whence 


a  'A 

X,   ' 


TJ  If 

1~>,    _     Al 

Ht~  K, 


(0 


Now  suppose  that  Z>,  and  Z>a  are  similar  ships  and  ul  and 
?/,  are  corresponding  speeds.  Then  from  §  61  (i)  we  have,  by 
the  law  of  comparison, 


A'«,' 


H,  =  DW 


(2; 


or 


Hence,  comparing  (i)  with  (2),  it  follows  that  K,  •*•  A\  =  i 
',  =  A',.     We  therefore  derive  the  important  result  that — 


A" 


34-6  RESISTANCE  AND   PROPULSION   OF  SHIPS. 

Similar  ships  at  corresponding  speeds  have  the  same  Ad- 
miralty coefficient. 

This  may  be  taken  as  a  statement  of  the  law  of  extended 
comparison,  and  as  such  it  will  be  equally  correct  with  any  of 
the  other  statements  implied  in  the  general  equation  of 
§  61,  (i).  This  statement  supplies  therefore  the  principle 
necessary  to  guide  in  the  choice  of  Admiralty  coefficients,  and 
onoe  such  principle  is  recognized  and  followed,  this  formula 
is  instantly  raised  from  its  former  condition  of  relative  unre- 
liability and  placed  on  the  same  level,  or  rather  made  iden- 
tical with  the  law  of  extended  comparison,  as  already  so 
numerously  expressed  above. 

As  an  illustrative  example  let  us  take  the  following: 

Dt  =  8000 ; 
«,=  18. 

Among  the  available  data  is  a  speed-power  curve  for  a 
similar  ship  of  displacement  6000.  We  have  now  to  find  the 
speed  at  which  the  latter  ship  will  correspond  with  the  former. 
Without  involving  any  other  data  we  may  take  the  ratio 
as  (D^-^-D^  =  1.049.  Hence  u,  =  18  -f-  1.049—  17.16. 
Turning  now  to  the  given  curve  at  a  speed  of  17.16,  suppose 
we  find  H  =  6320.  We  have  then  Dl  —  6000,  ul  —  17.16, 
and  HI  —  6320.  Substituting  this  in  the  Admiralty  formula 
we  find  K  =  264.  Now  by  the  law  of  comparison  as  above 
shown,  this  coefficient  is  suitable  for  the  similar  ship  of  dis- 
placement 8000  at  the  corresponding  speed  18.  Hence, 
making  the  substitution  in  the  formula,  we  find  for  the  pro- 
posed case 


POWERING   SHIPS.  347 


THE  ADMIRALTY  MIDSHIP-SECTION  COEFFICIENT. 


In  addition  to  the  Admiralty  displacement  formula,  the 
equation 

(area  of  midship  section)^1 
~KT 

has  been  used  to  a  considerable  extent,  especially  in  con- 
tinental Europe.  Where  the  value  of  Kl  is  selected  by 
judgment  alone,  it  stands  on  the  same  basis  as  the  D*  formula, 
though  it  is  hardly  as  satisfactory,  since  the  length  of  the 
ship  is  entirely  unrepresented  in  the  formula,  except  as  its 
value  is  taken  into  account  in  the  judgment  by  which  the 
value  of  Kl  is  selected.  Comparing  it,  however,  with  §  61 
(i),  it  is  seen  to  be  identical  with  the  D*  formula  in  its  rela- 
tion to  the  law  of  extended  comparison.  If,  therefore,  it  is 
used  in  this  way  for  similar  ships  at  corresponding  speeds,  it 
becomes  simply  another  mode  of  application  of  this  extended 
law.  We  may  thus  say,  as  before: 

Similar  skips  at  corresponding  speeds  have  the  same  midship- 
section  power  coefficient. 

•14-.  FORMULA  INVOLVING  WETTED  SURFACE. 

Formulae  have  been  proposed  and  used  quite  extensively 
of  the  general  form 


(•) 


As  proposed,  these  formulae  were  to  be  used  in  a  manner 
similar  to  that  in  which  the  two  Admiralty  formulae  have 
usually  been  used.  That  is,  the  values  of  K^  were  to  be 
selected  by  judgment.  They  are,  when  thus  used,  open  to 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 

the  same  objections  and  are  under  the  same  lack  of  confi- 
dence as  the  two  above  described  when  used  in  the  same  way. 

In  the  method  known  as  "  Kirk's  Analysis  "  an  approxi- 
mation to  the  wetted  surface  was  found  by  computing  that  of 
a  substituted  block  model  with  prismatic  middle  body  and 
wedge-shaped  fore  and  after  bodies.  The  appropriate  value 
of  K^  was  then  assumed  from  the  data  available  in  similar 
cases.  This  method  was  more  especially  intended  for  the 
powering  of  cargo  steamers  of  moderate  size  and  speed,  with 
an  abundance  of  data  available  to  serve  as  a  guide  in  the 
selection  of  the  values  of  the  constant.  Under  these  circum- 
stances the  method  gave  fairly  satisfactory  results,  as  indeed 
would  any  of  the  other  methods  here  discussed  when  used  in 
the  same  way. 

In  Rankine's  "  augmented  surface"  method  the  wetted 
surface,  or  more  commonly  the  reduced  surface,  §  8,  was 
multiplied  by  a  factor  called  the  coefficient  of  augmentation. 
This  coefficient  is  a  function  of  the  form  of  the  ship  at  the 
bow,  or  more  specifically  of  the  angles  of  maximum  inclina- 
tion of  the  various  water-lines  at  the  bow.  The  area  thus 
obtained  was  called  the  "  augmented  surface.  '  Its  value 
was  used  for  5  in  (i)  above,  and  appropriate  values  of  K^ 
were  assumed  exactly  as  in  other  cases. 

The  derivation  of  this  formula  proceeded  on  the  assump- 
tion that  the  only  resistance  necessary  to  consider  was  the 
frictional,  or  at  least  that  the  resistance  might  be  considered 
as  varying  with  the  square  of  the  speed  and  hence  the  power 
as  the  cube  of  the  speed  ;  and  further,  that  the  influence  of  the 
form  on  the  velocity  of  gliding  or  the  relative  velocity  of  skin 
and  water  could  be  represented  by  this  function  of  the  angles 
of  entrance.  The  method  is  interesting  as  containing  an 


POWERING    SHIPS. 

attempt  to  represent  the  influence  of  the  form  at  the  bow. 
It  is  entirely  inadequate,  however,  to  properly  provide  for 
either  variation  of  form  in  general,  or  for  varying  speeds.  It 
is  in  fact  subject  to  exactly  the  same  limitations  and  errors  as 
the  other  formulae  here  discussed,  and  requires  the  same  kind 
of  judgment  for  the  proper  selection  of  the  coefficient  in- 
volved. 

The  relation  of  such  formulae  to  the  law  of  extended  com- 
parison is  seen  to  be  identical  with  that  of  the  two  previously 
discussed.  In  general  it  is  seen  that  they  are  all  special  cases 
of  the  general  term  in  which  the  ship  is  represented  by  an 
area  or  second-degree  function,  and  the  speed  has  the 
index  3.  If  therefore  they  are  used  as  expressions  of  the 
laws  of  extended  comparison  they  will  have  the  same  authority 
and  will  give  the  same  results  as  any  of  the  other  expressions 
previously  given.  All  such  formulae  may  therefore  be  made 
expressions  of  the  law  of  extended  comparison  by  the  simple 
rule  : 

Similar  skips  at  corresponding  speeds  ^vill  have  the  same 
coefficient  in  t lie  formula  for  power. 


65.  OTHER  SPECIAL  CONSTANTS. 

The  various  forms  for  the  law  of  extended  comparison  in 
§  6 1  show  that  a  large  variety  of  "  constants  "  or  coefficients 
might  be  found  which  would  be  the  same  for  similar  ships  at 
corresponding  speeds.  The  constants  considered  in  §§  62, 
'">3.  and  64  are  all  derived  from  the  function  /*;/*.  Of  the 
various  constants  which  might  be  found  in  a  similar  manner 
we  will  only  refer  to  those  derived  from  the  functions  Du  and 


350  RESISTANCE  AND    PROPULSION  OF  SHIPS. 


D^u*.     Denoting  these  by  Kz  and  K^  we  have  for  the  corre- 
sponding formulae  for  power 

Du 


and  H  — 


>rre- 


,  -r       Du 

whence         K  =  - 


and 


If  for  similar  ships  at  corresponding  speeds  the  same 
values  of  the  constant  Kz  be  taken,  or  the  same  values  of  the 
constant  K^  the  law  of  comparison  will  be  fulfilled  and  the 
formulae  may  thus  be  used  for  the  determination  of  power  by 
means  of  this  law  in  the  same  manner  as  with  those  discussed 
in  §§  62,  63,  and  64. 

We  will  now  briefly  discuss  the  use  of  these  various  con- 
stants in  cases  where  the  geometrical  similarity  is  not  com- 
plete. 

66.  THE  LAW  OF  COMPARISON  WHERE  THE  SHIPS  ARE 
NOT  EXACTLY  SIMILAR. 

Strictly  speaking,  the  law  of  comparison  has  no  significance 
where  the  ships  are  not  similar.  Where  they  are,  any  and 
all  of  the  various  expressions  for  corresponding  speeds,  for 
resistance,  and  for  power  will  give  exactly  the  same  results. 
When  the  similarity  is  not  perfect,  however,  the  various 
expressions  for  corresponding  speeds,  for  resistance  and  for 
power  will  not  give  the  same  results;  and  while  strictly  none 
are  applicable,  it  is  quite  evident  that  they  may  all  be  consid- 


POWERING   SHIPS. 


351 


as  approximations,  more  or  less  near  according  to  the 
nature  of  the  similarity  and  to  the  particular  expression  used. 
It  is  evident  therefore  that  in  such  cases,  and  hence  in  most 
cases  likely  to  arise  in  actual  practice,  it  is  not  altogether  a 
matter  of  indifference  what  expressions  are  used  for  the  rela- 
tions between  the  speeds  and  between  the  powers  in  the  two 
cases.  We  must  first  note  clearly  the  two  questions  involved: 
(a)  The  relation  between  the  powers;  and  (U)  the  relation 
between  the  speeds,  or  the  definition  of  "  corresponding 
speeds."  With  regard  to  the  latter,  we  are  free  to  take  any 
of  the  various  expressions  of  §  26,  (7)  and  (8).  Correspond- 
ing speeds  are  intended  to  be  those  for  which  the  liquid  sur- 
roundings of  the  two  similar  ships  shall  also  be  similar.  These 
depend  principally  on  the  wave-formation,  and  this  on  the 
length  of  the  ship  more  than  on  any  other  one  feature.  It 
therefore  seems  preferable  to  fix  the  corresponding  speeds  by 
the  ratio  of  the  lengths,  and  we  take  therefore 


For  the  power-ratio  we  will  discuss  the  three  functions 
Du,  D^if,  Z^tt4,  or  the  functions  which  give  rise  to  the  three 
coefficients  K^  K,  K^  as  above  derived. 

Taking  the  first,  its  use  is  equivalent  to  the  assumption 
that  for  similar  ships  at  corresponding  speeds  the  power  varies 
as  the  function  Dut  or  in  symbols 

H  ~  Du. 

Denoting  the  chief  dimensions  of  the  ship  by  L,  B,  //, 
the  block  coefficient  by  b,  and  remembering  that  u  ~  L*  we 
have  for  similar  ships  at  corresponding  speeds 

BhLb 


R  ~~  D  ~  BhLb 


if  ~  Bhbu\ 


352  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

If  therefore  we  assume  more  broadly  that  resistance  in 
general  can  be  expressed  as  the  product  of  some  function  of 
the  ship  by  the  square  of  the  speed,  the  use  of  the  constant 
KS  is  equivalent  to  the  assumption  that  such  function  is  BJib. 

Similarly  we  have 

BhLb   4       Bhb 

R  ^  D  ~  BhLb  ~  —u* F—  u4. 

u  L 

Hence,  likewise,  if  we  assume  that  resistance  in  general 
can  be -expressed  as  the  product  of  some  function  of  the  ship 
by  the  fourth  power  of  the  speed,  the  use  of  the  constant  Kz 

is  equivalent  to  the  assumption  that  such  function  is  — -=. — . 

Now  taking  a  general  equation  of  resistance  involving  the 
second  and  fourth  powers  of  the  speed,  let  us  introduce  those 
functions  into  the  coefficients.  We  have  then 


The  equation  of  resistance  in  this  form  is  thus  seen  to- 
correspond  to  the  use  of  the  coefficient  Kz  as  above. 

In  a  similar  manner  we  find  for  the  form  of  the  general 
equation  which  corresponds  to  the  use  of  the  coefficient  K 
the  following: 

ID*\ 

R=P(D^+Q(-r  X (2) 

^  L,  I  . 

Likewise  for  that  which  corresponds  to  the  use  of  K^ 

R  =  P(DL)W  +  Q(Bhb]^ (3) 

But  in  §  8  we  have  seen  that  approximately  (DL)^  ~ 
wetted  surface  or  (DLJ*  ~  5.  Whence  we  may  write  (3)  in 
the  form 

R  =  P(S)u*  +  Q(Bhbyu* (4) 


POWERING   SHIPS.  353 

We  may  also  show  the  relation  of  the  three  functions  to 
the  dimensions  of  the  ship  and  the  block  coefficient  as  follows: 

For  comparison  the  function  Du  is  equivalent  to  L*Bkb  ; 


From  these  last  expressions  it  is  readily  seen  how  the 
different  values  would  vary  in  cases  where  the  similarity  is  not 
exact.  Thus  suppose  that  the  second  ship  is  relatively  longer 
than  the  first.  Then  the  length-ratio  will  be  greater  than  the 
others,  and  the  power  as  derived  from  the  above  expressions 
will  increase  in  amount  from  the  first  to  the  last.  Vice  versa, 
if  the  second  ship  were  broader,  deeper,  or  fuller,  the  first  of 
the  above  functions  would  give  the  largest  value,  and  the 
others  successively  less. 

Looked  at  from  the  standpoint  of  the  formula  of  resist- 
ance, it  would  appear  as  though  (3),  and  hence  the  function 
D^u*,  or  the  constant  K^  should  be  the  most  nearly  correct. 
Experience  seems  to  indicate,  however,  that  there  is  no  one 
form  equally  applicable  to  all  modes  of  variation  from  exact 
similarity,  and  sufficient  data  are  lacking  for  the  definite  rela- 
tion of  the  appropriate  form  of  function  to  the  nature  of  the 
departure  from  similarity. 

Mention  may  also  be  made  of  the  function  Z#,  which  is 
sometimes  used,  especially  in  connection  with  a  speed-ratio 

(A  -  A)». 

Of  these  various  functions,  D*u*  and  D*u*  or  the  constants 
K  and  A"4  may  be  recommended  in  preference  to  the  others. 
The  latter  is  perhaps  likely  to  be  the  more  correct,  though 
the  former,  from  the  fact  that  it  is  the  well-known  Admiralty 
coefficient,  has  been  much  used;  and  such  use  is  likely  to  con- 


354  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

tinue  so  long  as  no  more  definite  information  is  available 
regarding  the  relative  values  of  the  various  functions  and 
coefficients  available. 

We   may  illustrate  the   use   of  these   expressions  by  an 
example  as  follows.     Given  the  following  data: 


(I) 

(2) 400 


B 

h 

D 

u 

H 

40 

2O 

3700 

15 

312 

48 

24 

7640 

Then  U-;   =(1.33)*=  1.155. 

x^i 

Hence  «f  =  15  X  i.i$5  =  17-32. 

We  will  now  find  the  value  of  H^  using  the  following 
ratios,  the  results  being  as  shown : 

TT  TT 

"  a 

^ 


/z>  \« 

(jf-)  =  2.32  7260 

=  2.497  7814 


z£  =  2-385         7464 


.«,/  ~  VA  '  Vr  7  ~"""v" 

Where  the  variation  from  similarity  is  marked,  the  method 
of  comparison,  of  course,  fails  entirely,  and  it  must  be 
remembered  that  its  reliability  rapidly  decreases  as  departures 
from  similarity  increase.  The  discussion  of  the  present  sec- 
tion is  therefore  not  intended  to  show  a  means  of  extending 


POWERING    SHIPS. 


355 


the  method  of  comparison  to  widely  dissimilar  forms,  but 
simply  to  show  what  forms  of  expression  are  most  likely  to 
take  rational  account  of  such  slight  variations  as  experience 
indicates  may  be  admitted  without  sensibly  affecting  the 
application  of  the  law  as  stated. 


67.  ENGLISH'S  MODE  OF  COMPARISON.* 

Given  two  similar  ships  of  displacement  Dl  and  Z>2  at  speeds 
V^  and  Vv  not  in  general  "  corresponding."  Then  it  is  always 
possible  to  make  two  models  of  different  dimension  such  that 
at  the  same  speed  one  shall  correspond  to  Dl  and  the  other 
to  A-  From  the  principles  of  §  26  we  should  then  have  the 
following: 


Residual 
Resistance. 


Skin- 
resistance. 


Dis- 
placement. 

A 
A 


Speed. 


v 


Let  n  be  the  ratio  of  the  total  resistances  of  the  models 


at  the  speed  V\--     .     Then 


and 
Also 


•   —  TF  — 
,  -    W.T,, 


*  Proceedings  Institution  of  Mechanical  Engineers,  1896,  p.  79. 


356  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

Substituting  for  these  values  as  above,  we  find 


Then 

Total  resistance  of  Z>2  =  52  +  Wv 

The  values  of  Slt  Sa,  slt  st,  are  to  be  computed  in  the 
usual  way,  using  appropriate  values  for  the  skin-resistance 
coefficient.  Assuming  also  for  the  ship  Dl  the  ratio  between 
thrust  horse-power  and  I.H.P.  (CE  -=-  AE,  Fig.  66),  we  can 
find  5,  -f-  Wlt  and  hence  W^  The  value  of  n  is  found  by 
towing  the  models  </,  and  d*  simultaneously  from  the  ends  of 
a  bar  supported  at  an  intermediate  point.  The  point  of  sup- 
port is  then  adjusted  until  the  bar  remains  at  right  angles  to 
the  direction  of  motion.  In  this  condition  the  resistances  are 
evidently  in  the  inverse  ratio  of  the  lever-arms,  and  hence  a 
measurement  of  the  latter  will  give  the  value  of  n  as  desired. 
In  this  way  it  becomes  possible  to  determine  the  value  of  W^ 
above,  and  hence  of  the  total  resistance  52  +  W^. 

As  an  extension  of  the  above  method  of  treatment,  let  us 
assume  the  law  of  comparison  to  cover  the  total  resistance, 
instead  of  the  residual  only.  Let  the  experiment  be  made  as 
before  and  the  value  of  n  similarly  determined.  Then  denot- 
ing total  resistances  by  R»  /?„,  r,,  rv  we  have 


A  A 

<  =  "' * 


POWERING   SHIPS. 


357 


Hence 


or 


and  therefore 


. 

-—  =  ratio  of  powers  =  n[  jj 
-"i  * 


This  method  being  wholly  comparative,  no  dynamometric 
apparatus  is  necessary,  and  the  experimental  determinations 
are  comparatively  simple.  Since,  however,  the  results  are 
comparative  and  not  absolute,  its  field  of  usefulness  will  be 
much  restricted  in  comparison  with  the  usual  method  as 
described  in  §  26. 

68.  THE  VARIOUS  SECTIONS  OF  A  DISPLACEMENT  SPEED 
POWER  SURFACE. 

Let  us  consider  a  general  series  of  ships,  all  of  the  same 
type  of  form.  The  law  of  extended  comparison  furnishes  the 
connecting  law  between  all  the  ships  of  such  a  series,  no 
matter  what  the  displacement  or  speed.  It  is  therefore  evi- 
dent that  any  speed-power  curve  for  a  given  displacement 
may  be  considered  as  the  speed-power  curve  for  the  whole 
series  by  means  of  an  appropriate  modification  of  horizontal 
and  vertical  scales.  Suppose,  for  example,  that  we  have  a 
curve  for  displacement  Dl  plotted  with  scales  as  follows: 
Abscissa,  I  linear  unit  =p  knots.  Ordinate,  I  linear  unit  = 
q  I.H.P.  Then  the  same  curve  will  also  represent  the  speed- 
power  relation  for  D^  with  the  following  scales: 


Abscissa,   I  linear  unit  =  p\-^-j  knots; 

ID  \i 

Ordinate,  I  linear  unit  =  g\jr)  I.H.P. 


358 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


With  such  scale  modifications,  therefore,  one  curve  would 
represent  the  whole  series  of  values  of  power  for  the  various 
values  of  displacement  and  speed. 


8000 


6000 


4000 


2000 

DC 
Ul 

I 


10000 
8000 
6000 
4000 
2000 


10 


11 


14 


15 


15 


12  13 

*•   SPEED 

FlG.  95. — Relation  between  Speed  and  Power.     Displacement  constant  for 
each  curve,  as  shown  on  the  right. 


8000 


6000 


4000 


2000 

£T 
U 

1 


4000  6000 

DISPLACEMENT 


8000 


10,000 


on*     fr.r 


FIG.  96. — Relation  between  Displacement  and  Power.     Speed  constant  for 
each  curve,  as  shown  on  the  right. 

With  a  uniform  scale  such  a  series  of  values  would  require 
for    their    complete    representation    a    surface    of  which  the 


POWERING    SHIPS. 


359 


ordinate  is  the  value  of  //,  located  by  abscissae  corresponding 
to  the  given  values  of  D  and  u.  The  various  sections  of  this 
surface  by  planes  parallel  to  the  axes  of  H  and  u  would  repre- 
sent each  a  speed-power  curve  for  a  given  value  of  D. 

The  general  values  may  therefore  be  represented  by  a 
single  curve  with  varying  scales,  or  with  uniform  scales  and 
varying  curves. 

The  surface  above  referred  to  may  also  be  cut  by  planes 
parallel  to  the  axes  of  H  and  D  or  D  and  u,  thus  cutting  out 

16  [— 


ViQQQ 


Li  MX.) 


TOGO 


5000 


tooo 


3000 


3000 


8000 


10000 


4000  6000 

DISPLACEMENT 

FIG.  97. — Relation  between  Displacement  and  Speed.     Power  constant  for 
each  curve,  as  shown  on  the  right. 

curves,  one  set  showing  the  relation  between  H  and  D  for  a 
fixed  value  of  u,  and  the  other  the  relation  between  D  and  u 
for  a  fixed  value  of  H.  The  various  curves  of  section  are 
shown  in  Figs.  95,  96,  and  97,  a  study  of  the  general  charac- 
teristics of  which  may  be  recommended. 


69.  APPLICATION  OF  THE  LAW  OF  COMPARISON  TO  SHOW 
THE  GENERAL  RELATION  BETWEEN  SIZE  AND 
CARRYING  CAPACITY  FOR  A  GIVEN  SPEED. 

In  Fig.  98,  let  AB  denote  the  curve  of  E.H.P.  for  a  given 
ship   of   5000  tons   displacement   at   speeds   from    10  to  20 


RESISTANCE  AND   PROPULSION  OF  SHIPS. 

knots.  Now  suppose  this  model  taken  as  the  type  for  a  series 
of  vessels  of  displacement  from  5000  to  20  ooo  tons.  Then  by 
the  appropriate  transformation  the  E.H.P.  for  each  of  these 
vessels  may  be  determined  from  the  given  curve,  as  explained 
in  §  68.  Again,  assuming  a  constant  propulsive  coefficient  of 


JOOOO 
8000 
6000 
4000 

2000 
oc 

UJ 

2 

i 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

~jr 

^ 

/ 

^ 

s* 

c^ 

-< 

^-^ 

^ 

*^ 

^-  —  - 
A 

****'^' 

0        11        12       13       14       15       16       17       18       19       2C 

SPEED  IN   KNOTS 

FIG.  98. 

say  .54,  the  values  of  the  I.H.P.  may  be  similarly  obtained 
from  a  curve  CD,  the  ordinates  of  which  are  equal  to  those 
of  AB -^  .  54.  If  a  constant  propulsive  coefficient  could  be 
assumed  for  the  5OOO-type  ship,  CD  would  be  its  curve  of 
I.H.P.  Actually,  however,  with  the  same  machinery  at 
varying  speeds,  the  propulsive  coefficient  will  vary  so  that  CD 
would  not  in  such  case  correspond  to  an  experimental  curve 
derived  from  any  one  ship,  and  we  may  more  properly  con- 
sider it  simply  as  a  curve  proportional  to  E.H.P.  Let  us 
now  find  the  I.H.P.  for  our  series  of  similar  ships,  all  at  20 


POWERING   SHIPS. 


knots  speed.     The   speed   at  which  the   5OOO-ton   ship  will 
correspond  is 

/fOOO\i 


U  =  20 


D    I  ' 


Taking  the  ordinate  of  CD  at  this  speed,  let  it  be  denoted 
by  y.  Then  the  I.H.P.  for  the  similar  ship  at  20  knots  will 
be 

H  =  y 


500 

In  this  way  we  derive  AB,   Fig.   99,   which  shows  the 
I.H.P.  at  this  speed  for  the  supposed  series  of  ships.     The 


21 


CO 

16 


"I 

Q 

10 1 
*5 


0         Z        4         6         8        10        13       U        16       18       30 

DISPLACEMENT  IN  1000  TON  UNITS 
FIG.  99. 

form  of  the  curve  shows  plainly  that  H  increases  at  a  much 
slower  rate  than  D\  in  fact  in  this  particular  case  the  rate  is 


32  RESISTANCE  AND   PROPULSION  OF  SHIPS. 

nearly  as  the  square  root  of  the  displacement.  Next  suppose 
the  weight  of  the  hull  and  fittings  to  be  .50  of  the  total  dis- 
placement for  the  entire  series.  This  ratio  will  vary  consid- 
erably according  to  the  type  of  structure,  and  would  not, 
moreover,  remain  the  same  in  ships  of  so  great  difference  in 
size.  In  any  case,  however,  the  variation  will  not  be  suffi- 
cient to  affect  the  general  nature  of  the  relations  which  we 
wish  to  establish,  and  some  such  assumption  is  necessary  in 
order  to  introduce  the  necessary  simplification  into  the  rela- 
tions involved.  The  ordinates  to  the  line  OD,  Fig.  99,  will 
then  equal  the  displacement,  and  those  between  OH  and  OD 
the  weight  of  the  hull,  etc.  Taking  the  total  weight  of 
machinery  and  boilers  as  one-ninth  ton  per  I.H.P.  or  9 
I.H.P.  per  ton,  and  coal  at  1.8  pounds  per  I.H.P.  per  hour 
for  all  purposes,  and  assuming  enough  coal  for  a  seven-days* 
run  and  neglecting  the  variation  in  D  due  to  the  consumption 
of  coal  and  stores,  or,  otherwise,  considering  that  the  mean 
displacement  for  the  trip  corresponds  to  the  values  of  the 
diagram,  it  is  readily  found  that  the  weight  of  machinery  and 
coal  will  be  represented  by  the  ordinate  between  OH  and  CC. 
The  remainder  lying  between  CC  and  the  axis  of  X  is 
evidently  cargo-carrying  or  earning  capacity.  It  appears  that 
in  the  case  chosen  the  5OOO-ton  ship  could  barely  make  the 
trip  without  cargo,  while  as  D  increases  the  earning  capacity 
is  seen  to  rapidly  increase,  rising  to  3000  tons  forZ>  —  16  ooo 
and  to  4200  for  D  =  20  ooo. 

Again,  if  we  take  it  roughly  that  the  operating  expenses 
will  vary  with  the  power,  the  time  being  the  same,  the 
diagram  shows  how  rapidly,  relative  to  expenses,  the  earning 
capacity  increases  as  D  is  increased.  Thus  for  D  =  9000 
the  expenses  are  represented  by  14  400  and  the  earning 


POWERING   SHIPS. 


363 


capacity  by  about  1000  tons,  while  by  an  increase  of  D  to 
16000  or  by  77  per  cent  the  expenses  are  increased  by 
about  45  per  cent  and  the  earning  capacity  by  200  per  cent. 

Again,  in  Fig.  100,  let  us  note  the  variation  of  carrying 
capacity  with  speed,   the  length  of  voyage  being  constant. 


18,000 


16,000 


14,000 


12,000 


10,000 


JB.OOO 


6,000 


4,000 


2,000 


10          11         12          13          U         15         16          17 
SPEED  IN  KNOTS 

FIG.  100. 


18          19 


We  will  take  for  illustration  the  12  ooo-ton  ship.  AB  is  the 
curve  of  I.H.P.  The  constant  ordinate  between  CD  and  EF 
denotes  the  constant  weight  of  hull  and  fittings.  The  ordi- 
nates  between  EF  and  GH  denote  the  weight  of  machinery 
and  coal,  the  latter  decreasing  with  the  decrease  in  power,  but 
increasing  with  the  increase  of  time.  The  ordinates  between 
CD  and  GH  give,  then,  the  total  weights  on  the  same  suppo- 
sitions as  before.  The  remaining  ordinates  between  GH  and 
A' show  the  variation  of  carrying  capacity  with  speed,  and  how 
dearly  the  latter  is  bought  at  the  expense  of  the  former. 

The  particular  values  for  these  various  relationships  in  any 


364 


RESISTANCE  AND    PROPULSION  OF  SHIPS. 


given  case  will  depend  on  many  considerations  here  omitted. 
The  general  nature  of  the  results,  however,  is  sufficiently 
indicated  by  the  diagrams,  which  are  chiefly  intended  as  sug- 
gestive applications  of  the  law  of  comparison  to  the  study  of 
special  problems  of  this  character. 


I 


CHAPTER   VI. 
TRIAL  TRIPS. 

70.  INTRODUCTORY. 

THE  general  purpose  of  a  trial  trip  is  to  determine  the 
speed  and  power  which  may  be  maintained  continuously  for 
a  certain  distance  or  time.  In  addition  to  this  fundamental 
purpose  it  is  always  desirable  to  observe  such  data  as  bear  in 
any  way  on  the  general  problem  of  resistance  and  propulsion, 
while  in  particular  cases  various  characteristics  may  be 
observed  relating  to  different  points  on  which  information  is 
desired. 

For  the  direct  determination  of  speed  it  is  sufficient  to 
obtain  simultaneous  observations  of  distance  and  time.  We 
shall  not  here  consider  in  detail  the  measurement  of  power, 
assuming  it  to  be  determined  by  indicators  in  the  usual 
manner. 

The  details  of  the  various  measurements  necessary  for  the 
determination  of  speed  may  vary  somewhat  with  the  length 
of  the  course.  We  may  have  (i)  a  long  course,  as  for  exam- 
ple from  20  to  100  miles  or  more,  over  which  but  one  run,  or 
at  most  but  one  run  in  each  direction,  is  to  be  made;  or  (2)  a 
short  course,  as  of  one  or  two  miles,  usually  the  former,  over 
which  as  many  runs  may  be  made  as  desired. 

We  may  also  distinguish  between  the  purpose  of  the  trials 
as  follows:  (a)  the  determination  of  the  speed  or  power,  or 

365 


366  RESISTANCE  AND   PROPULSION  Of  SHIPS. 

both,  under  a  single  set  of  conditions,  such  as  full  power  or 
half-power  for  example ;  or  (b)  the  determination  of  the  speed 
and  power  under  a  series  of  varying  conditions  so  that  the 
continuous  relation  between  speed  and  power  for  widely  vary- 
ing values  of  either  may  be  determined. 

We  may  next  briefly  mention  the  chief  points  relating  to 
the  ship  which  may  influence  speed,  and  the  conditions  to  be 
fulfilled  for  maximum  speed. 

(a)  Displacement. — This  should  be  as  light   as   possible, 
though  this  consideration  is  not  independent  of  (b)  and  (c). 

(b)  Propeller. — The    surfaces    should    be    smooth,    blades 
thin,  edges  sharp,  and  it  must  be  well  immersed. 

(c)  Trim. — The  resistance  will  not  presumably  vary  sen- 
sibly for  slight  changes  of  trim,  but  it  may  result  that  a  very 
considerable  change,  such  for  example  as  might  be  necessary 
to  immerse  a  propeller  when  the  displacement  is  very  light, 
would   materially  increase  the  resistance  for  the   given   dis- 
placement.     In  such  case  there  is  also  the  added  loss  due  to 
the  obliquity  of  the  thrust. 

(d)  Bottom. — This  should  be  smooth  and  freshly  painted. 

(e)  Wind  and  Sea. — The  wind  should  be  light,  and  prefer- 
ably off  the  beam  or  a  little  ahead,  in  order  to  give  the  benefit 
of  increased  natural  draft  for  the  boilers.     The  sea  should,  of 
course,  be  smooth. 

For  the  attainment  of  the  fundamental  and  secondary 
purposes  of  trial  trips  the  following  data  are  requisite  or 
desirable : 

Power. 

Distance. 

Time. 

Revolutions. 


TRIAL    TRIPS.  367 

All  conditions  affecting  the  relation  of  the  ship  to  resist- 
ance and  propulsion,  such  as — 

Mean  draft. 

Trim  at  rest. 

Condition  of  bottom  if  known. 

Wave  profile  alongside  of  vessel. 

Change  of  trim  when  under  way. 

Depth  of  water. 

State  and  direction  of  wind  and  waves. 
In  addition  to  these,  which  are  observed  at  the  time  of  the 
trial,  the  characteristics  of  the  ship  and  of  the  propeller  are 
supposed  to  be  known. 

171.  DETAILED  CONSIDERATION  OF  THE  OBSERVATIONS  TO 

BE  MADE. 

Distance. — Omitting  special  reference  to  power,  we  first 
note  that  the  distance  with  which  we  are  concerned  is  that 
which  the  propeller  has  driven  the  ship  through  the  water, 
and  not  the  distance  over  the  ground  or  between  fixed  points 
on  shore.  The  influence  of  tides  and  currents  must,  therefore, 
be  eliminated  before  the  distance  actually  traversed  through 
the  water  can  be  known. 

For  the  marking  of  distance  itself  we  have  two  general 
methods — buoys  or  ships  at  anchor,  and  range-marks.  The 
former  are  more  suitable  for  a  long  course,  where  the  change 
in  location  caused  by  swinging  to  the  tide  will  be  of  no  relative 
importance.  For  a  short  course  such  errors  would  be  inad- 
missible, and  the  limits  of  the  course  must  be  marked  with  all 
possible  accuracy.  In  such  case  range-marks  on  shore  are 
made  lise  of.  Thus  in  Fig.  101  AB  and  CD  are  two  pairs  of 


368 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


poles  or  range-marks  determining  the  two  parallel  lines  AB 
and  CD.  The  course  is  then  some  line  EF  at  right  angles  to 
these  ranges,  and  at  such  a  distance  from  the  shore  that  the 
depth  of  water  shall  be  sufficient  to  avoid  sensible  retardation 
(§  19).  In  addition  to  the  ranges,  two  requisites  are  there- 
fore necessary  in  order  to  definitely  fix  the  course — a  direction 
and  a  location.  The  direction  is  usually  known  by  compass 
course,  and  it  is  also  desirable  to  mark  it  by  ranges  and 
buoys.  The  latter  means  is  also  used  to  indicate  its  approxi- 

TD 


FIG.  loi. 

mate  location  with  reference  to  the  proper  depth  of  water. 
At  each  end  of  the  course  there  should  be  plenty  of  room  for 
making  turns  and  for  gaining  headway  before  entering  the  run 
proper.  The  considerations  developed  in  §  77  show  that  the 
length  of  this  preliminary  run  should  be  preferably  not  less 
than  about  one  mile.  This  seems  to  be  in  agreement  with 
experience,  as  Mr.  Archibald  Denny  suggests  *  that  the  ship 
should  have  a  straight  run  of  at  least  one-half  mile,  and 
better  still  one  mile,  before  entering  the  course  proper. 


*  Transactions  International  Congress  of  Marine   Engineers  and  Naval 
Architects  in  Chicago,  1893.      Paper  xxxi. 


TRIAL     TRIPS.  369 

The  necessity  of  a  course  at  right  angles  to  AB  and  CD 
rather  than  on  some  oblique  line  as  E^  is  evident  without 
special  discussion. 

Tidal  Observations. — As  will  be  shown  later,  the  tidal 
influence  for  a  short-course  trial  may  be  approximately  elimi- 
nated by  appropriate  treatment  of  the  other  data  observed. 
With  a  long-course  trial  in  one  direction  only  such  elimina- 
tion is  not  possible,  and  with  a  single  run  in  each  direction 
the  data  cannot  be  satisfactorily  used  without  special  treat- 
ment for  the  elimination  of  tidal  influence.  It  is  therefore 
generally  desirable  to  have  made  special  tidal  observations. 
For  a  long  course,  boats  or  ships  may  be  anchored  at  intervals 
of  five  or  ten  miles  along  the  course,  and  should  make  observa- 
tions continually  as  often  as  once  in  fifteen  minutes  throughout 
the  trial.  Such  observations  should  preferably  be  made  at 
about  the  half  mean  draft  of  the  vessel  undergoing  trial.  The 
observation  may  give  simply  the  component  of  the  tide  in 
the  direction  of  the  course,  or  perhaps  preferably  the  whole 
velocity  and  its  direction.  Such  observations  are  usually 
made  with  a  so-called  patent  or  tarlrail  log.  This  consists 
essentially  of  a  small  screw  propeller,  which  in  the  case  of 
vessels  in  motion  is  towed  astern,  the  revolutions  being  com- 
municated through  the  towing  cord  or  by  other  means,  to  a 
counter  on  the  ship.  In  the  case  of  a  vessel  at  anchor  near 
the  line  of  the  course,  such  a  log  is  buoyed  a  short  distance 
from  the  ship  at  the  appropriate  depth,  and  observations- 
made  as  usual.  If  the  component  of  the  tide  in  the  direction 
of  the  course  only  is  desired,  the  log  must  be  maintained  in 
this  direction.  If  the  entire  velocity  and  the  direction  are 
both  to  be  observed,  the  log  must  have  a  vane  attached  and 
must  be  free  to  turn  to  the  tide.  As  shown  by  Froude's 


37°  RESISTANCE   AND    PROPULSION  OF  SHIPS. 

experiments,  such  instruments,  if  made  with  all  the  care 
•appertaining  to  the  production  of  a  piece  of  truly  scientific 
apparatus,  are  quite  reliable.  As  usually  furnished,  however, 
they  are  liable  to  a  correction,  and  should  be  rated  in  water 
known  to  be  still  at  a  speed  near  that  at  which  they  are  used. 
A  slight  error  in  the  log  is,  however,  of  less  importance  in 
tidal  observations  than  when  such  instruments  are  used  to 
determine  directly  the  velocity  of  the  ship  itself.  In  addition 
to  the  tidal  observations,  the  time  of  passage  of  the  ship  at 
each  station  is  also  noted. 

We  may  then  assume  that  we  have  thus  at  our  disposal  a 
mass  of  data  giving  at  from  four  to  eight  points  along  the 
course  a  series  of  tidal  observations  at  intervals  of  about  fif- 
teen minutes  for  the  whole  trial. 

For  a  short  course,  if  tidal  observations  are  made  at  all, 
the  number  of  locations  will  naturally  be  much  less,  according 
to  circumstances  and  the  degree  of  accuracy  with  which  the 
measure  of  this  disturbance  is  desired. 

Time. — For  the  measure  of  time  careful  observation  with 
accurate  watches  is  sufficient  for  long-course  trials.  For 
short-course  trials  the  probable  error  of  reading  with  the  usual 
form  of  seconds-hand  watch  may  become  sensible.  For  satis- 
factory accuracy  stop-watches  may  be  used,  or  preferably,  for 
scientific  purposes,  some  form  of  chronograph  by  which  an 
electric  signal  is  recorded  the  instant  contact  is  made  by  the 
pressure  of  a  key. 

Revolutions.  —  For  the  counting  of  revolutions  on  a  long- 
course  trial,  the  ordinary  engine-counter  is  quite  sufficient. 
For  a  short-course  trial  the  same  means  may  be  used,  though, 
as  with  time,  more  satisfactory  accuracy  is  attained  by  the 
use  of  some  chronographic  arrangement.  In  Weaver's  speed 


TRIAL     TRIPS. 


37* 


and  revolution  counter  a  paper  tape  is  fed  at  a  uniform  speed 
under  a  series  of  electrically  controlled  pens.  One  of  these 
under  control  of  a  clock  makes  a  mark  every  second.  Another 
is  in  circuit  with  a  contact  maker,  and  is  used  to  note  the 
instant  of  entering  and  leaving  the  course.  Other  pens  are 
electrically  connected  with  each  main  shaft  and  thus  register 
every  revolution.  Such  an  instrument  is,  of  course,  only  used 
in  short-course  trials. 

Special  Conditions. — Most  of  these  are  self-explanatory. 

The  wave  profile  may  be  determined  by  measuring  down 
from  the  rail  by  batten.  The  use  of  photography  from  a 
neighboring  vessel  may  also  be  suggested.  The  change  of 
trim  from  the  condition  at  rest  and  when  under  way  may  be 
determined  by  the  use  of  a  pendulum  adjusted  to  swing  in  a 
longitudinal  plane. 


72.  ELIMINATION  OF  TIDAL  INFLUENCE. 

We  first  suppose  the  observations  made  on  a  short-course 
trial.  These  may  be  considered  as  furnishing,  for  any  given 
run,  the  average  tidal  velocity,  and  the  correction  consists 
simply  in  subtracting  or  adding,  as  the  case  may  require. 

With  the  long  course,  where  several  hours  may  be  required 
to  traverse  its  length,  the  assumption  of  a  sensibly  uniform 
tide  is  not  admissible,  and  the  more  detailed  observations 
already  referred  to  become  necessary. 

An  approximate  method  of  applying  the  correction  is  as 
follows: 

Let  7*,,  Tv  7",,  etc.,  denote  the  lengths  in  minutes  of  the 
successive  time  intervals  between  passing  the  posts  of  obser- 
vation. Let  z>,,  v»  v»  etc.,  be  the  tidal  velocities  per  minute 


372 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


along  the  course  at  each  successive  station  at  the  instant  of 
passage  of  the  ship,  -f-  being  considered  as  denoting  a  velocity 
in  the  same  direction  as  the  ship.  Then  \(v^  -f-  z/J,  %(vy  -f- 
773),  -J(^3  +  vt)t  etc.,  are  taken  as  the  mean  tidal  velocities  for 
the  successive  intervals.  The  corresponding  tidal  influences 
will  be: 

For  the  first, 


second, 


third, 


-2J  =  ^  +  "2); 
7; 

T 

3/  I  \ 


etc.  etc. 

The  entire  tidal  correction  will  be  then  simply  the  alge- 
braic sum  of  these  partial  corrections. 

As  a  method  somewhat  more  accurate  in  principle,  the 
following  may  be  suggested: 

The  tidal  characteristics,  as  is  well  known,  are  expressible 
in  terms  of  circular  functions  of  time  and  distance  from  a 
given  origin,  and  the  tidal  velocity  at  the  ship  at  any  instant 
must  necessarily  vary  approximately  as  a  sinusoidal  function 
of  space  and  time.  In  Fig.  102  let  the  ordinates  at  (9,  P, 


FIG.  102. 


g,  etc.,  denote  the  tidal  velocities  at  the  successive  points  of 
observation  laid  off  on  a  time  abscissa.     Then  OP,  PQ,  etc., 


TRIAL    TRIPS. 


373 


are  the  successive  time  intervals  between  the  points  of  obser- 
vation. Now  without  attempting  any  refined  analysis  of  the 
tidal  components,  we  may  quite  closely  obtain  a  continuous 
value  of  the  tidal  velocity  at  the  ship  at  each  successive 
instant  by  passing  through  the  points^,  B,  C,  etc.,  a  smooth 
curve,  keeping  in  mind  the  characteristics  of  a  sinusoidal 
form.  If  now  the  area  of  this  curve  be  found  by  a  planimeter 
or  by  approximate  integration,  the  result  will  give  the  total 
distance  or  set  due  to  the  tide,  +  or  —  according  to  the  cir- 
cumstances of  the  case.  The  total  correction  found  in  either 
of  these  ways  is  then  applied  to  the  total  length  of  the  course, 
thus  giving  the  corrected  length,  from  which  in  conjunction 
with  the  time  the  speed  is  directly  found. 

We  will  next  consider  the  elimination  of  tidal  influence  for 
short-course  trials  without  special  observations. 

For  a  measured  mile  the  time  required  to  make  a  double 
run  with  and  against  the  tide  will  be  so  small  that  without 
large  error  the  tidal  velocity  may  be  considered  constant 
throughout.  In  such  case  the  simple  mean  of  the  two  runs 
will  evidently  give  the  true  mean  speed;  thus: 

(*,  +  v)  +  (u,  -  v)       u,  +  «, 


An  extension  of  this,  known  as  the  method  of  "  continued 
averages,"  has  also  been  much  used.  This  is  as  follows: 
Suppose  four  runs  made,  two  in  each  direction,  giving  appa- 
rent speeds  ft,,  fta,  ?/„  ?/4.  These  are  treated  as  shown  by  the 
form  on  the  next  page 

The  third  average  is  then  taken  as  the  correct  speed,  thus 
deduced  from  the  speeds  observed.  This  is  equivalent  to 
giving  the  second  and  third  runs  three  times  the  weight  of  the 


374 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


first  and  fourth,  and  the  final  result  may  be  more  readily 
found  by  this  formula  than  by  the  actual  formation  of  the 
successive  columns  of  averages. 


i1*7 


Speeds. 

ist  Averages. 

ad  Averages. 

3d  and  Final  Average. 

«! 

«2 

«3 
«4 

«i  +  «a 
2, 

«a  -f  «3 

Wi   -f-  2U<i  -f-  Us 

«i  +  3«a  +  3«3  -f  «* 

4 

«2  +  2«3   +   «4 

2 
«s  +  u4 

8 

4 

2 

The  correctness  of  this  method  depends  on  certain 
assumptions  which  we  may  briefly  note. 

Let  the  true  speed  remain  constant  at  u.  Let  the  tidal 
velocity  be  expressed  as  a  function  of  the  time  as  follows: 

v  =  a  +  bt  +  ct\ 

Let  the  runs  be  made  at  equal  time  intervals  which  may 
be  denoted  by  t.  Then  for  the  apparent  speeds  we  shall  have 


&a—  u  —  a  —  b  —  c\ 
U3  =  u  -\-  a  -f-  2b  -f-  4c ; 

By   substituting  in   the   formula  for  continued   averages 
above,  we  readily  find 


8 


=  u. 


It  thus  appears  that  if  the  true  speed  remains  constant, 
and  if  the  tidal  influence  varies  as  a  quadratic  function  of  the 


TKIAL    TRIPS. 


375 


time,  and  if  the  time  intervals  are  equal,  this  method  will  give 
the  true  speed.  In  the  actual  case,  however,  not  one  of 
these  conditions  will  in  general  be  fulfilled.  It  is  therefore 
doubtful  if  the  result  of  four  runs  thus  reduced  is  any  more 
accurate  than  the  simple  average,  or  than  the  average  of  two 
runs  made  with  as  small  a  time  interval  as  possible. 

It  may  also  be  noted  that  when  the  tide  is  just  on  the 
turn  either  way  the  water  is  frequently  affected  by  eddies 
and  counter-currents,  which  though  small  in  velocity  are  so 
confused  in  distribution  as  to  defy  any  attempt  at  elimination. 
When,  however,  the  tide  is  at  about  half-ebb  or  half-flow,  the 
velocity  is  greater  but  quite  regular.  This  results  naturally 
from  the  sinusoidal  character  of  tidal  motion,  as  a  result  of 
which  the  change  of  velocity  is  least  when  the  velocity  is 
greatest  at  about  half  tide,  and  the  change  is  greatest  when 
the  velocity  is  o  at  full  tide  or  slack  water.  It  follows  that 
half  tide  with  a  clearly  defined  regular  tidal  velocity  uniformly 
distributed  over  the  course  is  in  general  preferable  to  slack  or 
high  water  with  the  accompanying  eddies  and  irregularities. 

Further  methods  of  dealing  with  tidal  influences  will  be 
found  in  the  next  section. 

73.  SPEED  TRIALS  FOR  THE  PURPOSE  OF  OBTAINING  A  CON- 
TINUOUS RELATION  BETWEEN  SPEED,  REVOLUTIONS, 
AND  POWER. 

We  have  thus  far  been  concerned  with  the  actual  true 
speed  attained,  without  reference  to  the  corresponding  num- 
ber of  revolutions  or  the  I.H.P.  necessary.  For  the  investi- 
gation of  the  propulsive  performance,  however,  it  is  quite 
desirable  to  obtain  sufficient  data  to  furnish  a  continuous  rela- 
tion between  these  three  quantities.  For  this  purpose  the 


RESISTANCE  AND    PROPULSION   OF  SHIPS, 

short-course  trial  alone  is  suitable.  The  actual  conduct  of 
the  trial  is  in  no  wise  different  from  that  already  described, 
but  it  is  continued  through  a  decreasing  series  of  speeds  with 
their  appropriate  revolutions  and  powers,  to  a  number  of 
revolutions  as  low  as  can  be  maintained  uniformly  by  the 
engines.  The  problem  of  the  proper  disposition  of  the  data 
thus  found  is  more  complicated  than  when  speed  alone  is 
desired.  We  now  desire  to  determine  not  only  the  true 
speed,  but  also  the  revolutions  and  I.H.P.  which  correspond 
to  it. 

Now  in  practice  it  is  found  difficult,  even  for  anyone  run, 
to  maintain  the  revolutions  and  power  constant.  With  care 
the  variation  may  be  slight,  but  at  the  best  the  question  may 
always  arise  as  to  the  proper  interpretation  of  the  data 
observed. 

Let  //i,  Nlt  ul  be  the  power,  revolutions,  and  true  speed 
for  the  first  run,  and  //,,  7Va,  u.2  those  for  the  second,  v  being 
the  tidal  influence  taken  as  constant  for  the  two  runs.  Then 
the  actual  observations  furnish  us  with 

Ht,     N,,     (»,  +  »); 
fft,     ff,,     (»,-*). 

For  two  runs  intended  to  be  at  the  same  speed,  the  varia- 
tion will  be  so  slight  that  we  may  properly  assume  the  slip  of 
the  propeller  to  be  the  same  for  both.  As  a  result  the  true 
speed  will  vary  directly  with  the  revolutions,  and  the  true 
mean  speed  will  correspond  to  the  mean  number  of  revolu- 
tions. That  is, 


*.  +  »,  , 
corresponds  to 


TRIAL    TRIPS. 

Now  with  power  we  have  in  general 
H=Bun. 


377 


(0 


where  B  and  n  depend  alike  on  the  ship  and  on  the  speed. 
Where  the  variation  in  speed  is  not  large,  we  may  safely  take 
the  variation  in  B  and  n  as  negligible.  In  the  present  case, 
therefore,  we  should  have 


Whence 


flti 


and 


2Bn 


and 


.      .      (2) 


The  left-hand  member  of  this  equation  is  in  the  same 
general  form  as  (i)  above,  and  it  must  therefore  give  the 
value  of  the  power  which  corresponds  to  the  true  mean  speed. 
The  relation  of  this  to  the  observed  powers  Hl  and  //,  is  then 
shown  by  the  right-hand  side  of  (2)  above.  The  exponent  n 
will  usually  be  found  between  3  and  4.  As  an  illustration  let 

n  =  3 ; 
H,  =  3200; 
//,  =  3600. 


378  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

Then  by  substitution  we  find 


The  value  by  simple  average  would  be  3400,  and  as  the 
uncertainties  in  the  measurement  of  power  are  relatively  far 
greater  than  the  difference  thus  resulting,  and  as  these  values 
of  HI  and  H%  represent  a  variation  greater  than  would  be 
likely  to  occur  under  the  conditions  assumed,  it  seems  fair  to 
conclude  that  in  such  case  we  may  properly  take  the  mean 


REVOLUTIONS 

FIG.  103. 


power  as  corresponding  to  the  mean  speed.  For  two  runs 
thus  made  we  may  take,  therefore,  the  mean  revolutions, 
mean  speed,  and  mean  power,  as  all  corresponding. 

The   next   pair   of  runs   being   made  at  a  lower  power, 
another  set  of  means  is  found,  and  so  for  the  entire  series. 


TRIAL    TRIPS. 


379' 


Still  otherwise,  we  may  effect  the  general  averaging  by 
graphical  process  as  follows: 

Let  power  be  plotted  on  revolutions  as  in  Fig.  103,  each 
pair  of  spots  corresponding  to  a  double  run  intended  to  be 
made  at  the  same  power.  Then  a  fair  curve  passing  through 
and  between  the  spots  may  be  taken  as  the  best  approxi- 
mation to  the  continuous  relation  between  revolutions  and 
power.  We  may  then  plot  the  mean  of  the  speeds  as  an 
ordinate,  on  the  mean  of  the  revolutions  as  an  abscissa,  as  in 
OB,  Fig.  104,  thus  giving  a  continuous  relation  between 
speed  and  revolutions.  A  straight  line  OA  shows  similarly 


O  REVOLUTIONS 

FIG.  104. 

the  speed  if  the  apparent  slip  were  o.  Hence  the  vertical 
intercept  between  OA  and  OB  shows  the  corresponding  loss 
in  speed  at  each  point.  From  these  diagrams  we  may  then 
obtain  and  plot  if  desired  a  curve  showing  the  continuous 
relation  between  power  and  speed. 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 

We  may  also  derive  the  relation  between  revolutions  and 
speed  graphically  by  plotting  each  separate  speed  on  revolu- 
tions as  an  abscissa,  thus  obtaining  two  curves,  one  for  speeds 
with  and  the  other  for  speeds  against  the  tide,  with  pairs  of 
points  at  nearly  the  same  revolutions  on  each.  A  mean  curve 
will  then  give  the  desired  relation  between  revolutions  and 
speed. 

A  slightly  different  mode  of  graphically  determining  the 
relation  between  revolutions  and  true  speed  is  that  due  to 
Mr.  D.  W.  Taylor,*  which  is  briefly  as  follows: 

The  course  is  steamed  over  back  and  forth,  going  over 
each  time  the  same  route  off  as  well  as  on  the  course,  so  that 
the  time  intervals  between  the  middle  points  of  the  runs  may 
vary  inversely  as  the  speeds.  For  each  successive  run  the 
revolutions  are  decreased  by  as  nearly  as  possible  an  equal 
decrement.  This  gives  a  series  of  runs  at  certain  revolutions 
with  the  tide,  and  another  series  with  other  revolutions 
against  the  tide.  Each  of  these  is  then  plotted,  giving  curves 
of  speeds  plotted  on  revolutions,  with  and  against  the  tide. 
A  mean  curve  is  then  taken  as  the  curve  of  true  speed.  The 
author  then  shows  by  means  of  an  illustrative  example  that 
with  a  fulfilment  of  the  above  conditions  well  within  practi- 
cable limits  a  variable  tidal  influence  will  be  eliminated  with 
quite  sufficient  accuracy  for  all  practical  purposes. 

The  advantage  claimed  for  this  method  is  that  a  given 
range  of  speed  can  be  satisfactorily  covered  with  a  smaller 
number  of  runs  than  when  they  are  made  in  pairs  as  pre- 
viously described. 

*  Journal  Am.  Soc.  of  Naval  Engineers,  vol.  iv.  p.  587. 


TRIAL    TRIPS. 


381 


74.  RELATION  BETWEEN  SPEED  POWER  AND  REVOLUTIONS 
FOR  A  LONG-DISTANCE  TRIAL. 

While  the  intention  may  be  to  carry  on  such  a  trial  at 
uniform  power,  in  any  actual  case  there  will  naturally  arise 
considerable  variation  in  the  various  quantities  involved. 
The  mean  speed  and  the  mean  revolutions  are  readily  found, 
and  with  any  probable  amount  of  variation  we  may  presum- 
ably consider  them  as  corresponding  the  one  to  the  other. 
With  regard  to  the  power  which  corresponds  to  such  mean 
speed  or  revolutions,  however,  there  is  more  room  for  uncer- 
tainty. 

The  data  available  will  be  a  series  of  indicator-cards  taken 
at  intervals  throughout  the  trial,  and  giving  presumably  the 
mean  effective  pressures  in  the  cylinders  at  the  instant  in 
question.  The  revolutions  are  also  taken  at  the  same  time, 
and  thus  each  observation  gives  the  data  for  determining  a 
value  of  the  power  at  that  instant.  The  mean  of  these  revo- 
lutions, however,  will  not  be  the  true  mean,  due  to  individual 
errors  of  observation,  and  to  the  fact  that  they  are  merely  a 
series  of  detached  values  of  N,  while  the  true  mean  is 
obtained  from  the  continuous  counter  giving  the  total  number 
for  the  whole  run.  On  this  account,  therefore,  and  for  finding 
the  mean  power,  it  seems  preferable  to  use  the  mean  effective 
pressures  from  the  cards,  and  the  true  mean  revolutions  from 
the  total  counter.  The  method  for  doing  this  we  may 
develop  as  follows:  In  any  given  propeller  it  is  readily  seen 
that  the  relation  between  the  mean  effective  pressure  in  the 
cylinders,  or  more  definitely  between  the  mean  effective 
pressure  reduced  to  the  1.  p.  cylinder  (§  47),  and  the  thrust 
of  the  propeller  or  resistance  of  the  ship,  must  be  very  nearly 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 

constant  except  for  very  widely  varying  conditions.  We  may 
therefore  properly  assume  that  the  reduced  mean  effective 
pressure  will  vary  very  nearly  as  the  resistance.  Denoting 
these  pressures  by  /,,  />,,  etc.,  we  may,  with  all  significant 
accuracy,  assume  them  expressible  in  the  form  /,  =  bN*j 
etc.  Hence,  extending  the  method  outlined  in  §  73  for 
mean  power,  we  should  here  use  for  the  final  mean  pressure 
p  the  square  of  the  mean  square  roots  of  the  various  values 
A»  A>  etc-  This  pressure  used  with  the  true  mean  revolu- 
tions would  then  give  an  approximation  to  the  corresponding 
power  having  all  significant  accuracy,  at  least  so  far  as 
obtainable  from  the  data  at  hand. 

A  still  greater  refinement,  but  usually  without  practical 
value,  would  be  the  substitution  for  2  of  an  index  to  be 
derived  from  a  special  examination  of  the  relation  between 
speed  or  revolutions  and  power. 

If  the  variation  in  the  power  is  not  unusually  large,  the 
square  of  the  mean  square  root  will  not  sensibly  differ  from 
the  simple  arithmetical  mean,  and  in  such  case  the  latter  may 
properly  be  used. 

As  an  illustration  of  the  amount  of  the  variation,  let  three 
values  of/  be  denoted  by  64,  81,  and  100.  Then  the  square 
of  the  mean  square  root  is  8 1,  and- the  arithmetical  mean  is 
#1.66,  or  somewhat  less  than  one  per  cent  in  excess. 

We  therefore  conclude  that  in  all  usual  cases  we  may,  for 
the  power  corresponding  to  the  mean  revolutions  or  speed, 
use  the  simple  arithmetical  mean  of  the  pressures,  as  above, 
with  the  mean  revolutions.  At  the  same  time  the  substitu- 
tion of  the  square  of  the  mean  square  root  for  the  arithmeti- 
cal mean  will  undoubtedly  give  a  more  correct  result,  and  if 
the  difference  is  significant,  the  latter  method  should  be  use 


TRIAL    TRIPS. 


383 


75.  LONG-COURSE  TRIAL  WITH  STANDARDIZED  SCREW. 

We  have  thus  far  in  all  cases  assumed  the  measurement  of 
distance  to  be  effected  by  means  independent  of  the  ship's 
propeller.  A  method  of  long-course  trial  has  been  used  to 
some  extent,  however,  by  means  of  which  the  propeller 
itself,  having  been  standardized,  is  made  the  instrument  for 
the  measurement  of  the  distance.  This  is  known  as  the 
standardized  screw  method,  and  has  been  used  in  certain 
government  trials  at  the  suggestion  of  the  Bureau  of  Steam 
Engineering  of  the  Navy  Department. 

The  ship  is  first  taken  on  the  measured  mile  and  the  con- 
tinuous relation  between  speed  and  revolutions  as  discussed 
in  §  73  is  carefully  determined.  The  ship  being  in  the 
same  condition  as  regards  draft,  trim,  condition  of  bottom, 
etc.,  then  puts  to  sea  in  smooth  water  and  calm  weather,  and 
runs  such  a  distance  or  time  as  may  be  desired.  For  the  best 
accuracy  readings  of  the  counter  should  be  taken  every  ;ten  or 
fifteen  minutes.  In  any  event  the  total  time  and  total  Devolu- 
tions are  known.  Indicator-cards  are,  of  course,  taken  at 
appropriate  intervals  throughout  the  run. 

Now  if  the  revolutions  have  been  uniform,  or  if  they  have 
varied  somewhat,  but  not  out  of  a  range  within  which  the 
apparent  slip  may  be  taken  as  sensibly  constant,  then,  as  in 
§  73,  the  true  mean  speed  and  the  mean  revolutions  will 
correspond,  and  the  former  may  be  determined  directly 
from  the  latter  by  the  aid  of  the  curve  of  standardization.  If 
in  an  extreme  case  the  revolutions  are  widely  variable  and  the 
slip  is  widely  variable  with  revolutions,  this  correspondence 
will  not  be  accurate.  The  cause  of  the  error  and  a  method 
f  >r  its  elimination  may  be  seen  as  follows: 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


Let  OX,  Fig.  105,  be  a  time  abscissa  on  which  are  laid  off 
as  ordinates  the  successive  numbers  of  revolutions  per  minute 
at  the  instants  at  which  they  are  taken.  A  smooth  curve  AB 
drawn  through  the  points  thus  found  gives  a  continuous  rela- 
tion between  time  and  revolutions.  Then  from  the  curve  of 
standardization  the  speed  corresponding  to  each  number  of 
revolutions  may  be  found.  These  being  plotted  at  the  same 
points  will  give  a  curve  CD^  showing  the  continuous  relation 
between  time  and  speed.  The  area  OCDX  being  found  by 
planimeter  or  by  approximate  integration,  the  result  will  be 


FIG.  105. 

proportional  to  the  entire  distance,  and  the  time  being  known, 
the  true  mean  speed  is  thus  determined.  If  now  the  apparent 
slip  were  constant,  or  OB,  Fig.  104,  a  straight  line,  the  curves 
AB  and  CD  would  be  similar  and  the  integration  of  AB  would 
serve  the  same  purpose  as  that  of  CD.  But  the  integration 
of  AB  is  furnished  by  the  revolution  counter,  and  hence  the 
mean  revolutions  thus  determined  would  with  constant  slip 
give  the  true  distance  or  true  speed.  Again,  if  the  revolutions 


TRIAL    TRIPS. 


385 


are  constant,  it  is  evident  that  both  AB  and  CD  become 
straight  lines  parallel  to  OX,  and  their  constant  ratio  would 
be  that  between  speed  and  revolutions  given  by  Fig.  104,  for 
the  particular  value  of  the  latter. 

If  these  conditions  are  not  fulfilled,  that  is,  if  revolutions 
vary  with  time  and  slip  with  revolutions,  then  the  curves  AB 
and  CD  would  no  longer  be  similar,  and  the  use  of  the  former, 
or,  in  other  words,  of  the  mean  revolutions,  would  no  longer 
be  theoretically  exact. 

With  the  usual  amount  of  variation  in  these  quantities, 
however,  the  error  will  be  negligible  and  the  mean  revolutions 
may  be  used. 

The  treatment  of  the  power  has  already  been  considered 

in  §  73- 

As  some  of  the  advantages  of  this  method  mention  may 
be  made  of  the  following: 

(a)  There  are  no  tidal  corrections  to  consider.  This  arises 
from  the  fact  that  by  means  of  the  standardization  the  pro- 
peller is  made  the  instrument  for  measuring  distance. 

(/;)  The  ship  under 'trial  is  independent  of  the  presence  of 
other  vessels  for  course-markers  and  for  tidal  observations. 

(c)  The  results  are  not  invalidated  by  departure  from  a 
straight  course.     Any  course  desired  may  be  followed  so  long 
as  it  is  not  a  curve  of  small  enough  radius  to  give  rUe  to  a. 
sensible  retardation. 

(d)  The  results  attained  are  constantly  known  during  the 
trial,  so   that    tLe    exact   status   of   the  ship   relative   to   the. 
expected  performance  is  known  from  beginning  to  end. 

(e)  Any    derangement    to    the    machinery    which    might 

over  a  definite  course  not  only  inconclusive,  but 
•  1  ovoid  of  any  return  of  valuable  data,  will  have  no  effect 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 

on  the  result,  since  the  trial  may  be  considered  as  ended  at 
the  expiration  of  any  complete  ten-minute  interval,  and 
whether  the  results  are  considered  satisfactory  or  not,  at  least 
they  are  known,  and  valuable  data  are  obtained. 

On  the  other  hand  the  limitation  of  this  mode  of  trial  so 
far  as  relates  to  similarity  of  conditions  between  the  standard- 
ization trial  and  regular  trial  must  be  carefully  noted.  The 
apparent  slip  will  vary  with  the  condition  of  the  bottom,  and 
with  the  draft,  trim,  and  condition  of  the  wind  and  sea;  and 
since  calm  weather  and  a  smooth  sea  are  the  only  conditions 
which  can  be  definitely  described,  it  is  necessary  that  both 
trials  should  be  made  substantially  under  such  conditions. 

76.  THE  INFLUENCE  OF  ACCELERATION  AND  RETARDATION 
ON  TRIAL-TRIP  DATA. 

If  the  revolutions  and  power  or  mean  effective  pressure 
are  subject  to  rapid  fluctuations,  they  will  no  longer  have  the 
same  relation  to  speed  as  for  uniform  conditions.  If  the 
pressure  rapidly  increases,  a  portion  of  the  resultant  increase 
of  thrust  and  work  done  goes  toward  accelerating  the  motion 
of  the  ship,  and  the  mean  pressure,  revolutions,  and  power 
will  all  be  greater  than  for  the  existing  speed  under  uniform 
conditions.  Similarly  with  a  rapidly  decreasing  pressure  the 
mean  pressure,  revolutions,  and  power  will  be  less  than  for 
the  existing  speed  under  uniform  conditions. 

The  variation  of  the  pressure  is  usually  not  sufficiently 
rapid  to  make  this  item  of  practical  importance,  though  the 
possibility  of  its  existence  as  a  disturbing  feature  may  be 
borne  in  mind. 

It  could  only  be  eliminated  by  a  knowledge  of  the  vary- 


TRIAL     TRIPS. 


387 


ing  acceleration,  an  experimental  determination  of  which 
would  be  a  matter  of  no  small  difficulty.  A  further  problem 
involving  somewhat  similar  considerations  is  that  concerned 
with  the  time  and  distance  necessary  to  effect  a  change  of 
speed  either  of  increase  or  decrease.  This  is  of  such  general 
interest  and  importance  in  connection  with  the  subject- 
matter  of  the  present  chapter  that  we  may  properly  turn  to 
its  consideration  at  this  point. 


77.   THE  TIME  AND   DISTANCE   REQUIRED  TO  EFFECT  A 
CHANGE  OF  SPEED. 

Suppose  the  mean  effective  pressure  in  the  cylinders  to  -be 
dependent  alone  on  the  points  of  cut-off.  With  any  fixed 
condition  of  these,  therefore,  and  neglecting  variations  due  to 
friction,  the  engine-turning  moment  will  be  constant,  and 
hence  its  equivalent,  the  moment  resisting  the  transverse 
motion  of  the  propeller.  Again,  assuming  that  the  distribu- 
tion of  pressure  over  the  surface  of  the  propeller  is  sensibly 
the  same  for  constant  turning  moment  no  matter  what  the 
speed  of  the  ship,  it  follows  that  the  relation  between  the 
actual  thrust  and  the  mean  effective  pressure  is  geometrical, 
and  hence  if  the  former  remains  constant,  so  will  the  latter. 
While  these  suppositions  may  not  be  quite  exact,  they  must 
in  any  case  be  very  near  the  truth.  Suppose,  therefore,  the 
ship  being  at  rest,  that  steam  of  a  given  pressure  is  turned  on 
the  engine  with  the  cut-offs  in  such  position  that  their  joint 
result  would  give  a  mean  effective  pressure  sufficient  to  give 
a  thrust  which  would  maintain  the  ship  at  a  speed  of  u,  knots. 
In  consequence  the  ship  will  gradually  gather  headway  and 
after  a  time  attain  sensibly  the  speed  «,,  during  which  a  dis- 


388 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


tance  s  will  have  been  traversed.  In  thus  considering  thrust 
and  resistance  it  is  evident  that  actual  thrust  and  actual  or 
augmented  and  not  true  or  towed  resistance  must  be  taken. 

Let  R1  denote  the  resistance  at  speed  ^,  and  in  general  R 
that  at  speed  u.  We  shall  have  then  a  constant  thrust  equal 
to  R^  acting  against  a  variable  hydraulic  resistance  R.  The 
difference  Rl  —  R  will  be  a  net  force  available  for  the  accelera- 
tion of  the  ship.  We  must  also  remember  that  a  certain 
amount  of  water  is  more  or  less  closely  associated  with  the 
motion  of  the  ship,  and  partakes  of  its  acceleration  and  retar- 
dation. We  may  therefore  use  }*D  instead  of  D.  We  have 
then 


dt 


In  the  fundamental  formulae  of  dynamics  R  is  measured 
in  pounds  force  and  D  in  pounds  mass.  As  their  ratio  is 
unchanged  by  a  change  in  the  unit,  they  may  both  be  meas- 
ured in  tons.  We  have  therefore,  as  the  units  in  this  equa- 
tion, feet,  seconds,  ,.nd  tons.  Now  represent  R  in  the  gen- 
eral form 

R  =  cD*u\ (2) 

in  which  c  will  be  later  defined  so  that  R  and  D  may  be  in 
tons  and  u  in  feet  per  second  or  knots  per  hour.  In  this 
equation  c  is  not  necessarily  a  constant.  This  will  provide 
therefore  for  any  actual  law  of  augmented  resistance.  We 
have  then,  from  (i), 


TKIAL    TRIPS. 


389 


Dividing  by    ,    =  «,  we  have 


dt 


du 


Hence  from  (3)  and  (4) 


udu 

ds  =  - —  — j. 

gc  u,   —  IT 


(4) 


(5) 


(6) 


If  now  to  simplify  integration  we  assume  c  a  constant,  we 
have,  reckoning  t  and  s  from  starting, 

Ul  +  * 


2gCU, 


log. 


•  •     (7) 

•  -     (8) 


From  (7)  and  (8)  it  appears  that  when  u  =  «,,  both  t  and 
s  become  co  .  This  is  simply  a  result  of  the  form  of  the 
equations,  and  indicates  that  under  the  conditions  assumed, 
the  velocity  would  go  on  indefinitely  increasing,  but  would 
never  actually  attain  the  amount  ur  This  is  the  natural  and 

du 
necessary  result  of  the  decreasing  value  of  -7-  as  given  in  (3) 

when  u  approaches  «,,  and  while  it  may  be  a  suprising  result, 
it  is  in  exact  accord  with  the  physical  conditions  assumed.  In 
the  actual  case,  the  mean  effective  pressure  is  never  exactly 
constant,  and  hence  the  corresponding  velocity  ul  is  con- 
stantly variable  about  a  mean  value.  The  important  point 
practically  is  not  how  long  or  how  far  it  will  take  to  acquire 


390 


RESISTANCE   AND   PROPULSION  OF  SHIPS. 


the  velocity  ult  but  rather  a  velocity  sensibly  equal  to  it,  as 
for  example  .99^,,  or  .999^,.      These  relations  are  illustrated 

10,000 


15 
14 
13 
12 
11 
10 
""  9 

D     8 
U 

U 

fe    7 

6 
5 

4 


:<>7 


/ 

,v 


9,000 


8,000 


7,000 


6,000 


s 


5,000 


4,000 


3,000 


2,000 


1,000 


100 


500 


600 


200      300      400 

TIME  IN  SECONDS 
FIG.    106. — Time  and  Distance  for  gaining  Speed. 

in  Fig.  106,  where  values  of  (7)  and  (8)  are  plotted  for  succes^ 
sive  values  of  u.  We  will  now  explain  the  reduction  of  the 
equations  to  a  form  suitable  for  computation. 


TKIAL 


391 


Let  A'  be  the  Admiralty  displacement  coefficient  as  defined 
02,    and   e    the   ratio    between    the    thrust   horse-power 
1  .II.  P.  and  the  I.H.P.      Then  we  have 


I.H.P.  = 


and      T.H.P.  =  -~, 

A 


where  the  units  are  the  horse-power,  ton,  and  knot. 

To  reduce  to  the  units,  foot-ton  per  minute,  ton,  and  foot 
per  second,  we  have 

eD*u3  X  33000 


T.H.P.  = 


and        R  = 


x  2240 


KX  2240  X  (i.689)3' 

eD*u*  X  33QQQ  _  7.05  i*\     ,  , 

240  X  (1.689)"  X  6ou  ~  V    K   )Uu- 


Comparing  this  with  the  general  form  for  A*  in  (2),  it  is 
evident  that  c  =  .05  le  -=-  K.  This  may  be  taken  as  the  defin- 
ing equation  for  c  such  that  when  substituted  in  (2)  it  will  give 
R  in  tons  when  D  is  in  tons  and  n  in  feet  per  second.  Sub- 
stituting this  value  of  c  in  (7)  and  (8)  we  should  then  have 
the  values  of  /  in  seconds  and  s  in  feet,  expressed  in  terms  of 
D  in  tons  and  u  in  feet  per  second.  While  retaining  the 
second  and  foot  as  units  for  /  and  s,  it  will  be  more  conven- 
ient to  introduce  on  the  right-hand  side  of  the  equations  the 
necessary  factors,  so  that  common  logarithms  may  be  used 
id  of  hyperbolic,  and  speed  may  be  measured  in  knots 
id  of  feet  per  second.  Making  these  substitutions  in  (7) 
and  (8),  and  putting  /*  =  i.i,  we  find 


(sec.)  =  . 


, 
log 


(feet)  =  .  772  --  log 


u 


.(10) 


392  RESISTANCE   AND   PROPULSION   OF  SHIPS. 

For   illustration   take   K  —  240   and    K  ~  e  =  400,    D 
5000,  ut  =  15.     Then  we  have 


t  =  208.4  log 


2215 
and  s  =  5280  log  —  -f-^  .....  (12) 

The  numerical  values  of  (11)  and  (12)  are  plotted  in  Fig. 
106,  as  above  referred  to.  The  curve  AB  giving  the  time  is 
asymptotic  to  the  1  5-knot  line,  thus  showing  geometrically  the 
indefinite  approach  of  the  speed  to  the  15  knots  as  a  limit. 
The  curve  CD  shows  likewise  the  relation  between  time  and 
distance,  approaching  more  and  more  nearly  a  straight  line  in 
form  as  the  1  5-knot  speed  is  gradually  approximated  to. 

We  will  now  suppose  the  vessel  going  ahead  under  uniform 
conditions  at  speed  «„  and  that  the  engine  is  suddenly 
reversed  and  driven  with  full  mean  effective  pressure  astern. 
In  such  case  we  shall  have  a  different  geometrical  relation 
between  the  longitudinal  force  due  to  the  screw  and  the  mean 
effective  pressure,  than  for  motion  ahead.  In  general,  how- 
ever, we  shall  have  in  such  case 


du 

dt    '  '  ' 


where  P  =  backward  pull  due  to  engine,  and  R  =•  resistance 
to  velocity  of  ship  forward. 

We  will  now  assume,  similar  to  (i), 

R  =  c^u\ 
and  P  = 


TRIAL    TRIPS 


393 


where  for  P,  u,  is  the  speed  at  which  the  given  mean  pressure 
would  drive  the  ship  ahead.      We  have  then  for  (13) 

du  ?    . 


whence 


ds- 

g    t 
vtf 

:.«,*  +  ctf 
ndu 

.     .     .     (14) 
TiO 

Integrating  in  the  usual  manner,  we  have 


^-tan-^^-JJ;   .     (,6) 


('7) 


In  selecting  values  of  cl  and  c^  we  may  bear  in   mind  the 
following  considerations:   The  coefficient  <:„  is  less  than   the  c 
for  the  go-ahead  condition  due  to  the  rounded  backs  of  the 
blades,  and  the  consequent  decreased  longitudinal  component 
of  the  distributed  surface  pressures.      For  a  given  mean  pres- 
sure, therefore,  the  pull  would  be  less  than  the  thrust.     As 
to  the  ratio  between  the  two,  definite   information  is  lacking; 
and  in  default  of  such  definite  data  we  will  take  the  pull  as  .8 
the  thrust.      Again,  the  value  of  cl  corresponds  to  the  actual 
resistance  of  the  ship  to  motion  ahead  when  the  propeller  is 
backing.      In  this  condition   the  latter  is  sending  a  stream  of 
wati-r  forward  against  the  stern,  thereby  producing  an  excess 
of  pressure  at  this  point  instead  of  a  defect  as  when  working 
ahead.     The  actual  resistance  to  motion  ahead   is  therefore 


394 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


decreased  instead  of  increased,  or  the  augmentation  may  be 
considered  as  negative.  The  ratio  between  cl  and  the  c  for 
motion  ahead  at  the  same  speed  would  be  equal  therefore  to 
that  between  the  true  resistance  minus  this  decrement,  and 
the  true  resistance  plus  the  usual  augmentation.  Taking  the 
augmentation  and  decrement  about  the  same,  this  ratio  would 
be  not  far  from  .8,  and  we  will  therefore  take  c^  as  equal  to  c^ 
Comparing  also  with  the  value  of  c,  we  have 


Taking  /i  =  i.i,  substituting  these  various  values,  and 
reducing  speed  to  knots  and  hyperbolic  to  common  loga- 
rithms we  have 


s=  .959-^—^.30103  —  log^i  +  \^j  JJ.      .  (19) 
If  u  =  o,  we  have  simply 

<=-387irr: (20) 

.       .       .  (21) 


D-K 
s=  .289 =  .747*1*- 


With  the  same  numerical  values  as  before,  we  have 


(22) 


/  /       \  2\  ~H 

=  6560^.30103-  log  fl  +  ^jjj;         .       (23) 


and  if  u  =  o, 


=  177, 
—  1975- 


TRIAL    TRIPS. 


395 


The  varying  values  of  /  and  s  for  any  speed  u  are  given  in 
Fig.  107,  which  is  thus  a  diagram  of  the  extinction  of  forward 


2000 


1500 


^^ 

JUU 
190 
180 

160 

150 
140 

120s 

100  w 

90^ 

P 

80 

70 
60 
50 
40 
30 

10 
0 

'     B 

/ 

^ 

/ 

/ 

<,/ 

/° 

^ 

/ 

/ 

.p 

/ 

/ 

< 

o/ 

*/ 

/ 

c 

X 

/ 

o/ 

/ 

/ 

/ 

/ 

/ 

/ 

t 

/ 

/ 

/ 

1 

/ 

? 

/ 

> 

/ 

/ 

/ 

x 

/ 

/ 

/ 

/ 

/ 

7 

?i 

15       14       13       12       11       10 


8        7        6        5        4 

SPEED 


3        2 


TIME  AND  DISTANCE  FOR  STOPPING  FROM   FULL  SPEED 
FIG.  107. 

velocity.     AB  gives  the  relation  between  speed  and  distance, 
and  CD  that  between  speed  and  time. 

For  simplicity  of  operation  we  have  assumed   that   the 


RESISTANCE  AND   PROPULSION  OF  SHIPS. 

augmented  resistance  varies  throughout  as  the  square  of  the 
speed.  This,  of  course,  is  far  from  being  actually  the  case. 
The  ratio  K  -f-  e  may  be  expected  to  vary  considerably  in  the 
case  of  a  vessel  gaining  speed  from  rest  or  stopping  from  full 
speed.  If  the  law  of  variation  of  K  -^  e  were  known — that  is, 
the  law  of  the  variation  of  the  actual  resistance  with  speed, 
both  with  engines  turning  ahead  and  backing — the  funda- 
mental equations  could  readily  be  solved  by  approximate  or 
graphical  integration.  The  difference  in  the  result,  however, 
would  be  slight,  and  would  not  change  the  general  nature  of 
the  results  obtained  by  the  substitution  of  the  constant  value 
of  K  -r-  e.  It  may  be  remarked,  however,  that  in  the  selec- 
tion of  a  value  of  K  a  mean  rather  than  maximum  value 
should  be  taken. 

An  interesting  conclusion  from  (21)  is  that  the  distance  in 
which  a  vessel  may  be  brought  to  rest  from  motion  ahead  is 
sensibly  independent  of  the  speed,  and  depends  only  on  the 

j£ 

size  and  the  ratio  — ,  or  if  e  is  taken  as  practically  constant, 

on  D  and  K.  The  reason  why  there  must  actually  be  a  ten- 
dency toward  some  such  uniformity  in  the  distance  run  may 
readily  be  seen.  In  the  example  taken  this  distance  would 
be  from  five  to  six  lengths.  In  the  case  of  very  full  or  poorly 
formed  vessels,  or  with  foul  bottom,  we  might  have  an  average 
value  of  K  much  less  than  240.  As  an  illustration,  let  K  = 
1 80  and  K  -T-  e  =  300.  Then  the  distance  traversed  in  such 
case  would  be  only  three  fourths  of  that  under  previous  con- 
ditions, and  we  should  have  for  the  same  size  of  ship 

*=  133, 
s  =  1481. 


TRIAL    TRIPS. 


397 


These  figures  agree  with  the  general  results  of  the  experi- 
ments reported  to  the  British  Association,*  by  which  it  was 
found  that  ships  could  usually  be  stopped  in  from  4  to  6 
lengths,  and  that  this  distance  seemed  practically  independent 
of  the  power,  or  in  other  words,  of  the  speed  at  which  the 
experiment  was  made. 

We  have  assumed  in  all  the  foregoing  that  the  engine  is 
instantly  reversed,  and  that  the  engine  instantly  gathers  its 
motion  either  ahead  or  back,  as  the  case  may  be.  This  is  not 
actually  possible,  so  that  due  allowance  should  be  made  in 
comparing  experimental  results  with  those  given  by  the  fore- 
going approximate  theory. 

In  any  case  the  general  nature  of  these  results  is  correct, 
and  it  is  readily  seen  that  they  have  an  important  bearing  on 
the  manoeuvring  of  vessels  and  on  the  proper  conditions  for 
trial  trips.  In  measured-mile  trials  especially  the  importance 
is  clearly  shown  of  a  long  start  in  order  that  the  ship  may 
attain  sensibly  her  maximum  speed  before  entering  upon  the 
course. 


78.  THE  GEOMETRICAL  ANALYSIS  OF  TRIAL  DATA, 

Considering  the  three  items  of  data,  power,  speed,  and 
revolutions,  we  may  plot  the  former  on  either  of  the  latter  as 
abscissa  giving  curves  as  in  Figs.  108,  109,  or  we  may  plot 
speed  on  revolutions  giving  a  curve  as  in  Fig.  110.  The 
various  uses  of  these  curves  need  no  especial  explanation. 
In  using  the  power-speed  curve  for  purposes  of  comparison, 
§  61,  however,  it  must  be  remembered  that  the  given  curve 
strictly  applies  only  to  the  ship  in  the  given  condition  of  dis- 

*  Reports  for  1878,  p.  421. 


398 


RESISTANCE  AND    PROPULSION   OF  SHIPS. 


placement,  trim,  condition  of  bottom,   and  of  weather  and 
water,  and  that  it  cannot  be  applied  to  the  same  ship  for  all 


SPEED 
FIG.  108. 


REVOLUTIONS 

FIG.  109. 


^circumstances  without  due  allowance  having  been  made  for 
such  differences  in  condition  as  may  exist. 

It  is  frequently  of  interest  to  know  the  index  of  the  speed 
with  which  the  power  varies  at  any  given  speed,  and  similarly 


TRIAL    TRIPS. 


399 


for  the  variation  of  resistance  with  speed,  or  either  with  revo- 
lutions, or  in  fact  of  any  quantity  which  may  be  expressed  in 
the  general  form 

y  =  axn (i) 

For  example,   we  may  determine  that  at    10  knots  the 
power  varies  as  the  cube  of  the  speed  and  the  resistance  as 


u  REVOLUTIONS 

FIG.   no. 

the  square,  while  at  15  knots  the  power  may  vary  as  the 
index  3.7  and  the  resistance  therefore  as  the  index  2.7. 

We  will  now  develop  methods  of  deriving  the  value  of 
this  index. 

Let  x,  be  the  given  value  of  the  abscissa.  Then  the 
usual  method  has  been  to  assume  the  index  sensibly  constant 
for  a  short  range  on  either  side  of  this  value,  say  between  the 
values  x*  and  x^  equidistant  above  and  below  x^  Then  from 
the  general  equation  we  have 


40O  RESISTANCE   AND   PROPULSION   OF  SHIPS. 

whence  ^ 


and         „  =  log       -log       =  (2) 

0  J0  s  ;ro         log  *,  -  log  ;r0 

When,  however,  the  curve  is  at  hand  showing  the  relation 
between  y  and  x  as  is  usually  the  case,  the  following 
geometrical  method  will  be  found  much  more  rapid  and  satis- 
factory, besides  being  exact  in  principle  for  a  continually 
varying  value  of  n. 

We  must  first  obtain  a  clear  idea  of  the  nature  of  the 
index  with  which  we  are  here  concerned. 

Let  it  be  a  question  of  the  variation  of  any  function  y 
with  x.  Let  the  value  at  x0  bejv  Then  let  x  be  increased 
by  a  small  increment  such  that  its  new  value  divided  by  its 
old  is  (i  +  ^)»  where  e  is  a  quantity  very  small  compared 
to  I.  This  will  result  in  a  change  in  the  value  of  y  in 
a  ratio  which  we  may  denote  by  (i  +/).  Now  obviously 
we  may  put 


log  (i 

or  m  =  T  -        ......     (3) 

log  (i  +  e) 

This  we  take  as  defining  the  nature  of  the  exponent  m 
with  which  we  are  here  concerned.  It  would  therefore  follow 
that  if  m  —  3  and  the  speed  were  increased  i  per  cent  it 
would  result  in  an  increase  of  the  power  in  the  ratio  (i.oi)1. 

So  long  as  the  amount  of  increase  is  indefinite,  the  char- 
acter of  our  exponent  will  lack  mathematical  distinctness,  and 
we  are  therefore  naturally  led  to  define  m  as  the  ratio  of  log 
(i  -(-/)  to  log  (\-\-e)  when  e  is  indefinitely  decreased. 


TRIAL    TRIPS. 


4<DI 


Denoting  the  increments  by  dx  and  dy,  we  have  then  by  defi- 
nition 


**(<+?) 

By  Maclaurin's  theorem  of  expansion  this  becomes 

dy 


(4) 


We  have  also 


. 
dx    '    x 


(5) 


dx       d(\og  x)' 
x 


(6) 


We  must  now  show  that  the  exponent  m  thus  defined  is 
not  in  general  the  same  as  the  ;/.  of  the  general  equation 

y  =  axn. 

We  have  log  y  =  log  a  -\-  n  log  x. 

Now  remembering  that  in  the  cases  with  which  we  have 
to  deal  n  is  not  a  constant,  we  differentiate,  considering  both 
;/  and  x  as  variables.  We  thus  have 

dy  dn 

-  =  anx      -4-  axn  log  ^r-j-. 

dx  5    dx 


Also 


JK_ 
x 


whence  -  -r-  -  = 


dy  .  y  _ 

Tx^~x 


or 


w  =  *+  .r  log  ,r— 


402 


RESISTANCE  AND   PROPULSION   OF  SHIPS. 


Hence  it  follows  that  unless  n  is  constant,  m  and  n  will 
not  be  the  same.  It  also  follows  that  in  the  general  case  the 
exponent  found  by  the  expression  in  (2),  being  an  approxima- 
tion to  the  value  of  m,  will  not  satisfy  the  fundamental  equa- 
tion (i)  from  which  it  is  found,  except  for  a  particular  value 
of  a\  and  that  a  series  of  values  thus  found  will  not  in 
general  satisfy  the  fundamental  equation  (i)  for  any  constant 
value  of  a.  This  still  further  shows  the  fact  of  the  difference 
in  character  between  in  and  n  as  here  defined. 

Taking  then  the  value  of  m  as  thus  defined,  we  proceed  to 
show  a- method  for  its  geometrical  determination. 

In  Fig.  ill  let  OP  be  the  graphical  representation  of  the 


law  in  question.     Let  A  be  any  given  point  corresponding  to 
abscissa  OB,  and  let  CA  be  a  tangent  to  OP  at  A.     Then 


_  _ 

dx~  BC 


j    y    AB 

and         = 


TRIAL    TRIPS. 


403 


_  tly  ^  r  _  OB 

~~  d.v  *  x  "  BC 


(8) 


or  in  =  the  quotient  of  the  abscissa  by  the  subtangent. 

In  this  determination  the  only  uncertainty  is  that  involved 
in  drawing  a  tangent  to  a  curve.  This  may,  however,  be 
done  quite  accurately  as  follows:  We  may  evidently  assume 
without  sensible  error  that  the  curve  in  the  vicinity  of  A  may 
be  considered  as  parabolic  of  the  second  degree.  We  then 
take  two  points  E  and  F  equidistant  from  B  and  note  the 
corresponding  points  5  and  R  on  the  curve.  Then  by  a  well- 
known  property  of  such  curves  a  line  drawn  through  A 
parallel  to  RS  will  be  tangent  to  the  curve  at  A. 

The  exponent  n,  which  is  essentially  different  in  character 
from  ;;/,  may  be  defined  simply  as  the  exponent  which  will 
satisfy  the  given  fundamental  equation  y  =  ax".  Its  value  is 
then  most  readily  found  by  taking  logarithms,  whence 

—  log  a 


n  = 


(9) 


It  may  be  readily  seen  that  the  value  of  n  will  depend  on 
the  particular  unit  used  for  x,  but  once  this  unit  defined,  the 
value  of  n  becomes  definite.  Let  x  =  I,  or  the  unit.  Then 
we  have 

a=y, 

or  the  value  of  the  constant  a  =  the  ordinate  for  x  =  i. 
We  may  therefore  put  more  generally 

y  =  y>xn- 

The  exponent  ;//  may  be  seen  to  relate  simply  to  the 
nature  of  the  law  at  the  point  in  question.  We  may  therefore 
term  it  the  index  of  instantaneous  variation.  It  must  neces- 


404 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


sarily  be  the  exponent  involved  when  we  say  that  at  any 
point  y  varies  as  a  certain  power  of  x.  On  the  other  hand, 
the  exponent  n  depends  on  the  entire  law  of  variation  from 
the  point  where  x  =  I  to  the  point  in  question.  It  therefore 
represents  the  resultant  effect,  between  these  points,  of  the 


entire  series  of  values  of  m.  If  the  law  is  such  that  m  is  con- 
stant, m  and  n  become  identical  as  in  the  common  algebraic 
curves.  The  distinction  between  m  and  n  for  such  curves  as 
we  are  here  concerned  with  must  not  be  forgotten. 

The  application  of  the  preceding  to  the  variation  of  power 


TRIAL    TRIPS. 


405 


or  resistance  with  speed  is  obvious.  In  Fig.  112  are  given 
examples  in  which  AB  is  the  given  fundamental  curve,  CD  is 
the  locus  of  the  exponent  ///,  and  CE  is  that  of  the  exponent 
;/,  where  one  mile  is  the  unit.  These  diagrams  are  self- 
explanatory  and  will  repay  careful  examination.  Evidently 
the  same  investigation  may  be  made  with  reference  to  the 
power  and  revolution  curve.  The  speed  and  revolution  curve 
gives  the  apparent  slip,  which  may  also,  if  desired,  be  plotted 
separately  as  AB,  Fig.  113.  If  OA,  Fig.  104,  were  a  straight 


1-   dU 

i. 

^ 

JB 

SLIP  IN  PER 

o  S  5 

A 

— 

—  — 

-  —  • 

-—  — 

.  — 

-  —  • 

,—-- 

—  -" 

10                       *  15 

SPEED  IN  KNOTS 
FIG.  113. 

line  the  apparent  slip  would  be  constant,  AB  would  be  hori- 
zontal, and  the  indices  relating  power  to  revolutions  and 
speed  would  be  the  same.  Practically  this  is  never  found, 
though  frequently  the  slip  will  vary  but  slightly  through  a 
considerable  range. 

We  have  in  §  58  given  a  method  of  finding  the  mean 
effective  pressure  corresponding  to  the  initial  friction  of  the 
engine.  We  will  now  give  another  due  to  Wm.  Froude. 

In  Fig.  114  let  AB  be  a  curve  of  mean  pressure,  propor- 
tional, as  we  have  seen  in  £  47,  to  the  curve  of  indicated 


406 


RESISTANCE   AND    PROPULSION   OF  SHIPS. 


thrust.  Now  if  we  assume  that  this  pressure  varies  as  the 
resistance,  and  the  latter  for  speeds  below  uv  as  speed  with 
the  constant  index  ?/,  we  have  only  to  continue  BA  by  a 
curve  tangent  to  it  at  A  and  with  reference  to  some  at 
present  unknown  axis  EX,  fulfilling  the  condition  that  m  =  n 
at  all  points.  Froude  took  m  =  1.87,  and  the  construction 
is  readily  seen  to  be  an  immediate  result  of  the  general 
method  established  above  for  the  determination  of  m.  The 
construction  is  as  follows:  At  A  draw  a  tangent  to  the  curve.. 


SPEED 


FIG.  114. 


Lay  off  uf  —  u>O  -=-  m,  erect  a  perpendicular  CD,  and  draw 
EX.  Then  a  curve  EA  run  in  tangent  at  E  to  EX  and  at  A 
to  AD  will  approximately  fulfil  the  conditions  required  for 
the  continuation  of  the  pressure  curve  downward,  and  OE  will- 
therefore  be  the  value  of  />0,  or  the  value  for  the  initial  fric- 
tion. Due  to  the  uncertainties  of  drawing  a  tangent  at  the 
end  of  a  curve  and  the  uncertainty  that  the  pressure  varies; 
as  the  index  1.87,  or  indeed  with  any  other  constant  index  of 
the  speed,  the  final  result  necessarily  partakes  of  the  uncer- 


TKIAL    TRIPS.  407 

tainty   inherent   in   all   attempts   to   extend   a   graphical  law 
beyond  the  range  covered  by  the  observations. 

79.  APPLICATION  OF  LOGARITHMIC  CROSS-SECTION  PAPER. 

In  logarithmic  cross-section  paper,  the  divisions  are  similar 
to  those  on  a  slide-rule,  or  in  other  words,  the  distances  from 


ID 


4  A 


i_!_io 


i  Y.  \     / 


/ 


X 


/ 


/ 


02: 


4D     M-r, 


8        9      10 


FIG.  115. 


the  origin  are  proportional  to  the  logarithms  of  the  numbers, 
as  in  Fig.   115.      Cross-section   paper  ruled  in  this  way  h 


408  RESISTANCE   AND    PROPULSION   OF  SHIPS. 

many  special  applications  to  various  problems  in  naval  archi- 
tecture, that  a  brief  description  of  the  more  important  may 
be  of  interest. 

Given  an  equation  y  =  axH. 

Whence  log  y  =  log  a  -f-  n  log  x. 

Now  remembering  that  on  this  paper  the  abscissa  and 
ordinate  are  not  x  and  y,  but  log  x  and  log  y,  the  above 
equation  is  really 

ordinate  =  constant  -{-  n  times  abscissa. 
This  for  usual  plotting  is  in  the  form 


and  hence  represents  a  straight  line. 

It  follows  that  any  such  equation  will  on  this  paper  be 
represented  by  a  straight  line  inclined  to  X  at  an  angle  whose 
tangent  is  #,  and  cutting  Y  at  a  point  which  gives  a.  This 
fundamental  property  has  a  number  of  important  applications, 
of  which  we  may  note  the  following: 

(a)  If  a  =  I,  we  have 

y  =  xn. 

Hence  a  straight  line  on  this  paper  will  give  a  locus  of 
roots  or  powers,  of  index  whole  or  fractional,  to  determine 
which  it  is  simply  necessary  to  draw  a  line  or  a  series  of  lines 
at  an  angle  tan'1;/  to  X. 

As  commonly  made,  this  paper  has  but  a  single  logarith- 
mic scale  from  I  to  10,  subdivided  according  to  the  size  of 
the  unit.  By  drawing  on  this,  however,  a  number  of  lines 
dependent  on  the  nature  of  the  index,  a  single  sheet  may  be 
made  to  give  the  nth  power  or  root  of  any  number  from  o 

to  oo  .     For  illustration  we  will  take  the  case  of  a  table  of  f 

• 


TRIAL    TRIPS. 


409 


rers.  That  is,  we  wish  to  express  graphically  the  locus  of 
the  equation  y  =  ^r3.  Let  Fig.  1 16  denote  a  square  cross- 
sectioned  with  logarithmic  scales  as  described.  Suppose  that 


G 
FIG.  116. 

there  were  joined  to  it  and  to  each  other  on  the  right  and 
above,  an  indefinite  series  of  such  squares  similarly  divided. 
Then  considering  in  passing  from  one  square  to  an  adjacent 
one  to  the  right  or  above,  that  the  unit  becomes  of  the  next 
higher  order,  it  is  evident  that  such  a  series  of  squares  would, 
with  the  proper  variation  of  the  unit,  represent  all  values  of 
either  ^  or^  between  o  and  oo  . 

Suppose  next  the  original  square  divided  on  the  horizontal 
edge  into  2  parts  and  on  the  vertical  into  3,  the  points  of 
division  being  at  A,  B,  C,  E,  F,  G.  Then  lines  joining  these 
points  as  shown  in  the  diagram  will  be  at  an  inclination  to 
the  horizontal  whose  tangent  is  f .  Now  beginning  at  O,  OE 
will  give  the  values  of  .r*  for  values  of  JT  from  I  to  10.  For 
greater  values  of  x  the  line  would  run  into  the  next  adjacent 
square,  and  the  location  of  this  line,  if  continued,  may  be  seen 
to  be  exactly  similar  to  that  of  BC  in  the  square  before  us. 


4IO  RESISTANCE   AND    PROPULSION   OF  SHIPS 

It  follows  that,  considering  the  units  as  of  the  next  higher 
order,  the  line  BC  will  give  values  of  x*  for  x  between  O  and 
G  or  10  and  31.6+.  For  larger  values  of  x  we  should  run 
into  the  adjacent  square  above  with  change  of  unit  for/,  but 
without  change  for  x.  We  should  here  traverse  a  line  similar 
to  GF.  Therefore  by  proper  choice  of  units  we  may  use  GF 
for  values  of  x%  where  x  lies  between  31. 6 -|-  and  100.  We 
should  then  run  into  the  next  square  on  the  right  requiring 
the  unit  for  x  to  be  of  the  next  higher  order,  and  traverse  a 
line  similar  to  AD,  which  takes  us  finally  to  the  opposite 
corner  and  completes  the  cycle,  the  last  line  giving  us  values 
of  x*  for  x  between  100  and  1000.  Following  this,  the  same 
series  of  lines  would  result  for  numbers  of  succeeding  orders. 

A  little  consideration  of  the  subject  will  show  that  the 
value  of  x^  for  any  value  of  x  between  I  and  oo  may  thus  be 
read  approximately  from  one  or  another  of  these  lines;  and 
if  for  any  value  between  I  and  oo  ,  then  likewise  for  any  value 
between  o  and  i.  The  location  of  the  decimal  point  is  readily 
found  by  a  little  attention  to  the  numbers  involved.  A  rule 
for  its  location  might  be  derived,  but  is  of  little  additional 
value  in  practice.  The  limiting  values  of  x  for  any  given  line 
may  be  marked  on  it,  thus  enabling  a  proper  choice  to  be 
readily  made. 

The  principles  involved  in  this  case  may  be  readily  ex- 
tended to  any  other,  and  it  will  be  found  in  general  that 

if  the  exponent  be  represented  by  — ,   the  complete  set  of 

lines  may  be  drawn  by  dividing  one  side  of  the  square  into 
m  and  the  other  into  n  parts,  and  joining  the  points  of  divi- 
sion as  in  Fig.  1 16.  In  all  there  will  be  (m  +  n  —  i)  lines,  and 
opposite  to  any  point  on  X  there  will  be  n  lines  correspond- 


TRIAL 

the  n  different  beginnings  of  the  ;/th  root  of  the  ;;/th 
power,  while  opposite  to  any  point  on  Y  will  be  m  lines 
corresponding  to  the  ;;/  different  beginnings  of  the  wth  root 
of  the  wth  power.  Where  the  complete  number  of  lines 
would  be  quite  large,  it  is  usually  unnecessary  to  draw  them 
all,  and  the  number  may  be  limited  to  those  necessary  to 
cover  the  needed  range  in  the  values  of  x. 

If,  instead  of  the  equation  y  =  xn,  we  have  a  constant 
term  as  a  multiplier,  giving  an  equation  in  the  more  geneial 

m 

form  y  =  Bxn,  or  BxH,  there  will  be  the  same  number  of  lines 
and  at  the  same  inclination,  but  all  shifted  vertically  through 
a  distance  equal  to  log  B.  If,  therefore,  we  start  on  the  axis 
of  Y  at  the  point  B,  we  may  draw  in  the  same  series  of  lines 
and  in  <i  similar  manner. 

It  will  be  noted,  of  course,  that  the  index  ;;/  -f-  n  may  be 
used  either  way  for  the  same  set  of  lines.  That  is,  the  same 
lines  will 'give  the  £  or  f  power  of  numbers,  or  the  second  or 
£  power,  etc. 

(b]  If  in  the  general  equation  y  =  axn  the  exponent  n  is 
variable,  the  locus  plotted  on  logarithmic  paper  will  be  curved 
instead  of  straight,  and  the  index  of  instantaneous  variation 
m,  at  any  given  point  (§  78),  will  be  the  tangent  of  the  incli- 
nation of  the  tangent  line  at  such  point.     This  is  readily  seen 
from  §  78,  (6).     Also,  //  will  be  the  tangent  of  the  inclination 
of  a  line  drawn  from  the  point  where  the  locus  cuts  Y  (or 
where  x  =  i)  to  the  given  point.      This  readily  follows  from 
£  78,  (9).      This  serves  also  to  clearly  illustrate  the  difference 
in  character  between  ;;/  and  ;/. 

(c)  Proportions  of  the  form 


412  RESISTANCE  AND    PROPULSION  OF  SHIPS. 

are  frequently  met  with.     As  an  equation  this  is 


or  log  y^  —  log  yl  +  ;*(log  *\  —  log  *,). 
Now  in  Fig.  117  let  A  and  B  denote  the  points  x^  and  x^, 
and  £T  and  D  be  at  the  heights  denoted  by  yl  and  jj/9.  Then, 
remembering  that  the  actual  distances  involved  are  the 
logarithms  of  the  corresponding  quantities,  we  readily  see  that 
BD  =.  AC  +  nAB  or  AC  +  nCE.  Therefore,  a  line  from  C 
to  D  will  be  at  an  inclination  to  X  whose  tangent  is  n.  If, 


a?, 


FIG.  117. 

therefore,  we  have  any  ready  means  of  passing  from  C,  a 
point  determined  by  the  coordinates  xl ylt  along  an  oblique 
line  inclined  to  the  axis  of  X  at  an  angle  whose  tangent  is  #, 
it  is  evident  that  we  shall  pass  along  a  continuous  series  of 
points  x^y^  so  related  to  xlyl  that  the  proportion  above  men- 
tioned will  be  fulfilled.  To  solve  any  such  proportion,  there- 
fore, we  have  simply  to  start  at  the  given  point,  xly^  and 
pass  along  the  oblique  line  until  we  reach  a  point  whose 
abscissa  is  x^.  The  corresponding  ordinate  will  be  the  desired 


TRIAL    TRIPS. 


413 


value  of  ja.  Or,  vice  versa,  if  we  stop  at  any  given  value  of 
the  ordinate  jy,,  the  corresponding  abscissa  will  be  x^,  so 
related  to  the  other  quantities  that  the  proportion  is  fulfilled. 
To  provide  the  necessary  means  for  moving  in  the  right 
direction  from  any  point  whatever  as  ,r, y,,  a  series  of  equi- 
distant lines  at  the  proper  angle  may  be  ruled.  With  this  aid 
a  very  close  approximation  to  the  proper  values  of  x^y^  may 
be  made.  As  an  instance  of  the  use  of  this  proportion, 
suppose  that  we  wish  to  find  the  corresponding  speeds  for  two 
vessels  from  the  sixth  roots  of  the  displacements. 

We  have  —  =  \~ 

it,       \Dl 

Given  the  ul  anu  D1  we  find  the  point,  considering  one 
axis  as  that  of  speed  and  the  other  as  that  of  displacement. 
Then  by  passing  along  a  line  inclined  at  an  angle  tan"1  ^  to 
the  axis  of  D,  we  shall  pass  through  a  continuous  series  of 
points  which  will  give  corresponding  displacements  and 
speeds.  Stopping  at  the  displacement  D,  we  read  off  directly 
the  desired  speed  //„ ;  or,  vice  versa,  stopping  at  a  speed  «2  we 
read  off  the  corresponding  displacement  Z>2. 

Having  thus  found  the  corresponding  speeds,  we  may  by 
another  application  of  the  same  principle  find  the  I.H.P.  by 
using,  for  example,  the  equation 


As  a  further  illustration  we  may  take  the  determination  of 
Admiralty  coefficients  from  the  equation 

K=    -jj-. 

ill  of  the  operations  here  involved  may  be  carried  out  on 
logarithmic  paper  and  the  value  of  the  coefficients  rapidly 
determined. 


STEAMSHIP  AND    PROPELLER   DATA. 

IN  Tables  A  and  B  will  be  found  a  collection  of  data 
relating  to  ship  and  propeller  performance.  A  few  of  the 
cases  given  in  Table  B  are  not  represented  in  A  due  to  the 
difficulty  of  obtaining  satisfactory  values  of  some  of  the  data 
required.  In  Table  B  it  should  be  noted  that  the  value  of 
the  I.H.P.  is  that  required  for  one  propeller,  and  not  that 
required  for  the  ship  as  a  whole  in  the  case  of  twin  or  triple 
screws. 

The  data  presented  in  these  tables  have  been  gathered  from 
various  sources,  as  hear  as  possible  in  all  cases  to  the  origi- 
nal publication.  All  ratios  and  derivative  quantities  have, 
furthermore,  been  independently  computed  from  the  funda- 
mental data,  and  while  it  can  hardly  be  hoped  that  among  so 
many  figures  all  errors  have  been  avoided,  it  is  hoped  that 
their  number  is  small  and  their  character  not  significant.  The 
values  of  the  form  coefficient  i  for  the  propeller  have  been 
computed  from  the  developed  forms  of  the  blades,  and  show 
how  great  a  variation  may  exist  in  this  factor,  even  among 
propellers  which  might  by  general  estimation  be  considered 
the  same  in  form. 


In  Table  B,  column  headed  Screw ',  the  letters  have  the 
following  significance: 

C center 

S starboard 

P port 

M mean 

414 


TABLE   A, 


415 


f-2 


O^oooooo  Q  O  co  O  wo  NO  O  m  co  w  co  O  co  r^  r^  I-H  O  OO  O* 
\r>  +~  N  Q^OO  r^  «n  1-1  MOO  O  co  co  r^  O  in  -t-  1-1  in  co  O  OOO  co 
moo  mo  Lfit^-OOO  mo  u->o  in  m  ino  O  mt->-ooo  mm 

N  C^O  co  -1-  r^  N  O  O  coo  M  O  oo  co  oo  O  ~t-  O  f^O  mo  in  O  oo 
O  co  O^OO  moo  cnmQOOOO  mmmr^inoo  mr^o  •—  ca  M 

O^CXDOOOOOO    t^CO    O"CO    O^CX3COCOCO    I~>.OOOCOOOO     Q^OrD    r^^O^I^ 

rf-OOO  O*  -t  N  mO  m  rt  N  rj-  IH  rfOco  O*-i  COMO  r^W  M  r^ 
O^m—>-NQOOMOO<NOm  cnco  r^mNoo  O  NO  i-  NCO 

r^m'^-o  mco 

r^.  m 

mmcor^cnt^oo  ino^O  inmoo  i^mO  ^  w  t^mooco  »HCOOO 
mcomi-c  mr^cor^N  i-^d  e*  •-"  OO  W  N  N  co  ^-O  O  co  O^  en  >n 
mO'-iooQOP<OO^'TONcrv.r»r^T}-  MI-HTJ-  o^cor^Tt 

coor^Nr^^tooor^r^oomoromOOOr^OOOOQr^ 
•^-  o  o  moo  -to  i->  q  o  co  r^co  occT'-'oOTt^ 

M  r^rnc>Oi^   c>'^'^t-r;-t- 

882&&88888S 

O  cJo'coccJ  N  d  mo'  <N  Ttcococ  wcoo  '-  ^  i-  w  mro 
co  TJ-  m  rt  ^t  COO  Tj-  co  Tto  co  -3-  *1-  rr  M  mmco"-i  •^•coco-l-coi-i 

qqwqq»nincoqq>nqqqc>qoqqr^qqqqqq 

S*O    C^mmr~«»Hc5coc>O    OOmQvwM   cooo  oo   co  r>-  O   O   O   O 
r^mi-i   i-ioo   Pi   O^N   t^O   O^O   <-<   >-<    ^"-"O   N   cor^mrtw   MO 

CONaTt-NH. 


ac  c 

ils 


•S-c 


11 


MmOMONOOomOoor^comooONcomNcocoOOco 
mwco    OO   coi^i-O   comO   m'1-rj-comco   N   O   t^O   •-•   mo 
O^O    •^•comoo   coo   r^-a   1-1   O   mmrl- 
co  o  •*  ^       r^  r>.  IH  coco  *•  co  ^t  N 


416 


STEAMSHIP   AND    PROPELLER   DATA. 


O   Q 

11 

!<§ 

a   OO  <N  r*  *t  O 
O   O   O   -^  "~>  t^»  en\ 

in\o  vO  O  O   m  in 

^eno^fo   ^Oooo  r».vo   ^-CM  ent-ioooo   om 
r^^  ^"  co  o   ""J"  Tt~  ^r  O  M   o   O   O  O  r^  *o  in  o  O 

Qu  £    G 

Oco  r^  r*»  O  Ooo 

x>   o  Oco  t^oo  t^  r^co  o  r^co  r^co  co  co   Oco   o  O 

i^s 

ij 

M   Tj-co  co   O   *?f  m 
in  in  ^i-  -rfO   in  -^- 

mcM   enco   encooococo   OO   CNOOCO   CM    <~i   MOO   ^-m 

«« 
U 

Area 
Midship 
Section. 

iiKHl 

en 

o  eno^-w  o5  S 

81^-  r)-oo  vOOvnOoocoOcoOenOOooOenO 
co   OO*^O  co   o  GO  oo  t^  r^»  O  en  ^  o  O  m  ^  o 

2  w 

r^*  en  O^oo   O  co  co 

m  en  -^-oo  co  en-rj-^t—  enenM  CM  CM  TJ-M  ^tcM  rf-i- 

1? 

"1 

is 

8O   O   O   O   O   CM 
m  O   O   O  O   O 

8  in  enco  inOf^r^Oco   O   O   enenO   O^nO   ^t~Q 
NNOOOvOOOcovOOcoenoONOOO 

s 

|£ 

rt  m  en  CM   rf  en  -3- 

OO  i^'l-CM  inTj-^j-'rj-i-t  M  enencnen  eno   m  in  en 

< 

•s  • 

O  O  O  «o  O   O  >o 

O   O   O   O  mo   CM   N   Tj-oo   OOOOOMOOOO 

_] 
m 

tan 

O  cs  oo   rf  en  O  CM 
O   O  co   O  ^^  ***}  co 

mcoOOCMOOOcO'3-OOooOf^TtcocMOt~>' 

< 

V 

"H.  w  o 

OO     O  N    CM    M    -^-(X3 

en  cf°°  ^  S  J?  ^5- 

M 

en  O 

^  CM  *o   O  O  t^»  CM  en  co  CM  ^  vo  *n  o  in  o  *o  oo  o   o 
r^-O'-'en      COCMCMI^            M        MN        ot^^CM 

MM                                                                                                                        M 

4) 

1 

.....    oJ 

•:::•£ 

1    •    *    '•    '.  v 

~  :  :      :      :'.::'•::• 

~  B  a  1    ^1 

iillill 

"w  *  '•     *&     .'•'.'.     tj  '•  '.  t>- 

1  -        ?.s""  *~J>  *§J  rt^l^e 
SS       ^jjS.     «Il^?l88*I§ 

a«  g  M  o3.25-    ^  g^o  cU-e-CSg  S  c 
g  *°  ^  -S  S  3  rt       rtojo"3"5003"303-^ 

TABLE  A. 


417 


co  coco  coooo  N  N  oo  O  O 
co  oo  t->«oo  t^  to  o  r-»  ooo  co 
xr>  m  u->  in  m\o  in  o  in  in  in 


o  d  o  co  N 
\O   TJ-  OO   m 


rr  w  r>.  ^t  co  M  <3-  O  r~* 
O  r>.oo  -i  t*~  t^  t~*ao  o 
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STEAMSHIP  AND    PROPELLER    DATA. 


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TABLE   B. 


419 


HI  04  coo  04   Tt-  q  *toO(Oi   O  r^  r^.  O  tnco   Ooo  to  r^o  O  co  HI 

<     C- 

co  co  co  co  u~i  HH  HI  co  co  O  O  co  co  r*»  r^»  r^»  r^«  r*»  O  co  O  O  O  O  O 
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rt  •  o^wo4ooo4(^qo^<^qcocoaocococsr^OOOO 

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420 


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STEAMSHIP   AND    PKOPELLER   DATA. 


C«*-T3 


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I    UU  UPQ      ~ 


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TABLE   B. 


421 


<     ou 


rj-co   O^oO  *f~>  O  O  Ooo   M  r^O   MOO   m  co  co  O  oo   OOO   — 

1   1   *? 


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ur-»-Hco\o  ec 


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II 


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miniomminO    O   r^O    «    ioir)ir>ir) 


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422 


STEAMSHIP   AND    PROPELLER   DATA. 


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r^r^OOOOfOcococococooOO 


cocoO  M   cococococxjooooco 


NMTj-rJ-COCOCOCOCOCOCOCOCOCOCOCOCOCO-n-aWOJMCOCO 


TABLE   B. 


423 


coo»-|>-'CMCMONOt^r^ou"><OcQ   coco 

-'•n  v 

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inmo   O   u->ir>N 


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424 


STEAMSHIP   AND    PROPELLER    DATA. 


r-  r-  xn  i-«  co  T-  r~.<o   in  m  en  co  a     in  r-c     e    r-OO   rcc      ^  »H  in 


O  C^  Oco  co   en  en  O  co   CM   O   d   N   vnQ   O   *-' 
i^r^r^vOvOvOvOoo   oc>o  miomo  O  ^J 


TABLE  B. 


I-1    U^  COCO  OOCOOOOvniDM»-iMMO^U-)NC<vOO*-'CO 


r^Tj-vOQO  O  Ooo  O  •^••^-»r>vr>rfNOxno  t-oor^a 
M  oo   t>:  r^  w   coo   O>  •-  O  f*  coco  -too  ^  O  M   O>  O  co 

OOOOCOiDloa-CO'^COI^l-i'-ICOCO  OOO    O  l-i    W    a 


intoo        coo   O  coco 

aNWO*t-iNWl-iM 


coot-it-iowr-^r^r^NCJ 
O  O  C 


o  o  \  6  6  [  o  q  d  d  d  d  d  d 


is~>.  -23  S?,  ~  =  -«2 


INDEX. 


Acceleration  and  retardation,  influence  of,  on  trial  data 386 

Admiralty  displacement  coefficient 343 

Admiralty  midship-section  coefficient 347 

Air,  indraught  of 292- 

Air- resistance 125 

Analysis,  geometrical,  of  trial  data   397 

Analysis  of  power  required  for  propulsion 230 

Apparent  slip 2ifc 

Apparent  propeller  efficiency 228 

Area  factor,  diagram  for 253 

Area-ratio 186,  253, 

Augmentation  of  resistance  due  to  action  of  propeller 229 

Augmented  surface,  Rankine's 348 

Auxiliaries,  power  required  for 336 

Barnaby 185,  198,   298 

Beaufoy : 34,  40 

Bilge-keels,  influence  of,  on  resistance 123 

Blechynden 252,  286,  337 

Blenheim    H.M.S no 

Bureau  of  Steam  Engineering.  Navy  Department 383 

Biisley,  Carl 147 

Calvert 148,  209 

Cam  ere 115 

Canals,  resistance  in no 

Carrying  capacity,  size  and  speed,  relation  between 359 

Cavitation 295 

Change  of  speed,  time  and  distance  required  for 387 

Coefficient  of  augmentation 232,  234 

Colthurst 55 

Comparison,  law  of 128,   142 

Comparison,  law  of,  applied  to  propeller  design 300 

Comparison,  law  of,  when  ships  are  not  exactly  similar 350- 

Conder 115 

Corresponding  speeds 133 

Cotterill 39,  198,   248- 

427 


4-8  INDEX. 

PAGE 

Course  for  trial  trips 368 

Data 415,  425 

Dead-water 5 

Denny 368 

Derume 116 

Diameter-ratio 177 

Disk  area  defined 171 

Dubuat   34,   115 

Effective  horse-power 230 

Efficiency  defined 160 

Efficiency  of  screw  propeller 189,  192,  228 

Encyclopaedia  Britannica 56 

Encyclopaedia  Metripolitana 56 

Engine  friction 328 

English's  mode  of  comparison 355 

Euler 3,   151 

Experimental  data  connected  with  equations  for  propeller  design 243 

Experimental  methods  of  determining  resistance 148 

Feathering  paddle-wheels 202 

Foul  bottom,  influence  of,  on  resistance 55,  127 

Friction,  engine 328 

Froude,  R,  E 34,  79,  100,  180,  195,  197,  198,  243,  285 

Froude,  Wm 34,  50,  106,  123,  125,  149,  405 

Gatewood 56 

Greyhound I 126,   149 

Guide-blade  propellers 205 

Head-resistance  coefficients 34 

Helicoidal  area  defined 171 

Hichborn 55 

Hull  efficiency 231 

Hydraulic  propulsion 203 

Indiana,  U.S.S 319 

Indicated  thrust 239 

Indraught  of  air 292 

Initial  friction 329,  331 

Isherwood's  experiments 251 

Joessel's  experiments 34 

Kinematic  similitude,  law  of 128,  142 

Lagrange 3 

Load  friction 334 

Location  of  propellers 290 

Logarithmic  cross-section  paper,  application  of 407 

Mas,  M.  de 116 

Massachusetts,  U.S.S 319 

Materials  for  propeller-blades 308 


INDEX. 


429 


PAGE 

Maxwell 19 

McDermott 263 

Mean  pitch 211 

Mean  pitch,  computation  of 216 

Mean  slip  defined 211,  214 

Merkara 106 

Middendorf 146 

Negative  apparent  slip 240 

New  York,  U.S.S no 

Normand 296 

Number  of  propellers 288 

Obliquity   of  stream  and   of    shaft,   influence  of,  on  action  of   screw 

propeller 220 

Paddle-wheel,  action  of  element  of 165,  167 

Paddle-wheel,  propulsive  action  of 198 

Parsons 298 

Pitch-angle. .    172 

Pitch  defined 171 

Pitch,  measurement  of 320 

Pitch,  modification  of,  due  to  twisting  of  blade 323 

Pitch  ratio 172 

Pollard  and  Dudebout   56 

Powering  ships,  Chapter  V 326 

Powering  ships,  illustrative  example 338 

Powering  ships  by  law  of  comparison 340 

Powering  ships  by  formulae  involving  wetted  surface 3_!7 

Powering  ships  by  use  of  special  constants 349 

Propeller  design,  Chapter  IV 243 

Propeller  design,  equations  for 263,  264,  282.  283,  284 

Propeller  design,  problems 261 

Propeller  design,  tables 264,  265 

Propulsion,  Chapter  II 157 

Propulsive  coefficient 23° 

Propulsive  element,  action  of 159 

Projected  area  defined lll 

Racing 294 

Rankine 16,  5$,  198,  204,  348 

Reduced  wetted  surface 49»  5T 

Resistance,  Chapter  I l 

Resistance    air I25 

Resistance,  eddy 29 

Resistance  formulae r4^ 

Resistance,  head  and  tail 3° 

Resistance  in  canals no 

Resistance,  increase  of,  due  to  banks  and  shoals no 


43°  INDEX. 

PAGE 

Resistance,  increase  of,  due  to  change  of  trim 120 

Resistance,  increase  of,  due  to  irregular  movement 108 

Resistance,  increase  of,  due  to  rough  water 109 

Resistance,  increase  of,  due  to  slope  of  currents 119 

Resistance,  oblique,  of  ship-formed  bodies 151 

Resistance  of  planes  moving  normal  to  themselves 32 

Resistance  of  planes  moving  obliquely  to  their  normal ; . . .  .      37 

Resistance,  relation  of,  to  density  of  liquid 106 

Resistance,  residual 31,  144 

Resistance,  skin  or  tangential,  of  planes 40 

Resistance,  skin  or  tangential,  of  ships 49 

Resistance,  skin  or  tangential,  values  of  constants 52-55 

Resistance,  stream-line 28 

Resistance,  surface  or  skin 29 

Resistance,  wave-making 29,  95 

Risbec 47,  104 

Russell,  J.  Scott 56,  113,  115 

Sardegna,  bilge-keels  on < 124 

Screw  propeller,  action  of  element  of 172 

Screw  propeller,  definitions ,  „ . . 169 

Screw  propeller,  geometry  of 310 

Screw  propeller,  laying  down 314 

Screw  propeller,  propulsive  action  of , 177 

Screw  turbines 205 

Shallow- water  resistance no 

Shape  factor 256 

Shape  factor,  computation  of 257 

Sink,  stream  line 18 

Size,  carrying  capacity,  and  speed,  relation  between 359 

Slip  angle  defined 173 

Slip  angle,  table  for  connecting  with  slip  ratio 323 

Slip  defined 163,  211 

Source,  stream  line 18 

Speed,  size,  and  carrying  capacity,  relation  between 3:9 

Speed-trial  with  standardized  screw 3?3 

Springing  of  blades 299 

Stokes 56 

Si  ream -lines 7 

Stream  lines  defined 4 

Strength  of  propeller  blades 303 

Stopping,  distance  and  time  required  for 387,  395 

Surface  coefficient  diagram 253 

Surface  ratio 1 86,  253 

Sweet,  Elnathan 114 

Taylor,  D.  W 26,  39,  112,  147,  325,  380 


INDEX. 


Thornycroft 185 

Thornycroft's  screw  turbine 205 

Thrust  horse-power 230 

Tidal  influence,  elimination  of 371,  378,  580 

Tidal  observations 369 

Tidman 40 

Trial  trips,  Chapter  VI 365 

True  propeller  efficiency 228 

True  slip 212 

Units  of  measurement xi 

Wake,  constitution  of 208 

Wake  factor 227 

Wake,  influence  of,  on  equations  of  propeller  action 224 

Wake-return  factor 234 

Waves 56 

Waves,  combinations  of 65 

Wave-echoes,  formation  of 91 

Wave-formation  due  to  motion  of  ship 77 

Waves  of  translation 71 

Waves,  shallow- water 69 

Waves,  tabular  data 75~77 

Waves,  train  of,  propagation  of 89 

Wetted  surface 50,  51 

Wetted  surface,  reduced 49,  51 

Wetted  surface,  powering  by  formulae  involving 347 

White no,  1 20 

Yarrow 121,  149 


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World's  Columbian  Exposition  of  1893.  .4to,  half  morocco,  10  00 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  00 

Dyer's  Light  Artillery 12mo,  3  00 

Hoff' s  Naval  Tactics 8vo,  1  50 

Hunter's  Port  Charges Svo,  half  morocco,  13  00 

2 


Ingalls's  Ballistic  Tables 8vo,  $1  50 

"        Handbook  of  Problems  in  Direct  Fire 8vo,  4  00 

Malmn's  Advanced  Guard 18mo,  1  50 

Permanent  Fortifications.  (Mercur.).Svo,  half  morocco,  750 

Mercur's  Attack  of  Fortified  Places 12mo,  2  00 

Elements  of  the  Art  of  War 8vo,  400 

Metcalfe's  Ordnance  and  Gunnery 12iuo,  with  Atlas,  5  00 

Murray's  A  Manual  for  Courts-Martial .18mo,  morocco,  1  50 

"        Infantry  Drill  Regulations  adapted  to  the  Springfield 

Rifle,  Caliber  .45 18mo,  paper,  15 

Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner 12mo,  4  00 

Reed's  Signal  Service 50 

Sharpe's  Subsisting  Armies 18rno,  morocco,  1  50 

Very's  Navies  of  the  World 8vo,  half  morocco,  3  50 

Wheeler's  Siege  Operations 8vo,  2  00 

Winthrop's  Abridgment  of  Military  Law 12mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene 12mo,  1  50 

Young's  Simple  Elements  of  Navigation..  12mo,  morocco  flaps,  2  00 

first  edition 1  00 

ASSAYING. 

SMELTING — ORE  DRESSING— ALLOYS,  ETC. 

Fletcher's  Quaiit.  Assaying  with  the  Blowpipe..l2mo,  morocco,  1  50 

Furman's  Practical  Assaying 8vo,  3  00 

Kunhardt'sOre  Dressing 8vo,  1  50 

*  Mitchell's  Practical  Assaying.     (Crookes.) 8vo,  10  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  CO 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  00 

Thurston's  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Wilson's  Cyanide  Processes 12mo,  1  50 

The  Chlorination  Process 12mo,  150 

ASTRONOMY. 

PRACTICAL,  THEORETICAL,  AND  DESCRIPTIVE. 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Practical  Astronomy 8vo,  4  00 

Gore's  Elements  of  Geodesy 8vo,  2  50 

3 


Hay  ford's  Text-book  of  Geodetic  Astronomy 8vo. 

Michie  and  Harlow's  Practical  Astronomy 8vo,     $3  00 

White's  Theoretical  and  Descriptive  Astronomy 12ino,       2  00 

BOTANY. 

GARDENING  FOR  LADIES,  ETC. 

Baldwin's  Orchids  of  New  England 8vo,  1  50 

London's  Gardening  for  Ladies.     (Downing.) 12mo,  1  50 

Thome's  Structural  Botany 18mo,  2  25 

We^termaier's  General  Botany.     (Schneider.) 8vo,  2  00 

BRIDGES,  ROOFS,    Etc. 

CANTILEVER — DRAW — HIGHWAY — SUSPENSION. 
(See  also  ENGINEERING,  p.  6.) 

Boiler's  Highway  Bridges 8vo,  2  00 

*  "       The  Thames  River  Bridge 4to,  paper,  5  00 

Burr's  Stresses  in  Bridges 8vo,  3  50 

Crehore's  Mechanics  of  the  Girder 8vo,  5  00 

Dredge's  Thames  Bridges 7  parts,  per  part,  1  25 

Du  Bois's  Stresses  in  Framed  Structures 4to,  10  00 

Foster's  Wooden  Trestle  Bridges 4to,  5  00 

Greene's  Arches  in  Wood,  etc 8vo,  2  50 

Bridge  Trusses 8vo,  250 

RoofTrusses 8vo,  125 

Howe's  Treatise  on  Arches 8vo,  4  00 

Johnson's  Modern  Framed  Structures .4to,  10  00 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  I.,  Stresses 8vo,  2  50 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  II..  Graphic  Statics 8vo,  2  50 

Merrimau  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

Part  III.,  Bridge  Design ,8vo,  2  50 

Merriman  &  Jacoby's  Text- book  of  Roofs  and  Bridges. 

Part  IV.,  Continuous,  Draw,  Cantilever,  Suspension,  and 

Arched  Bridges 8vo,       2  50 

*  Morisou's  The  Memphis  Bridge Oblong  4to,     10  00 

4 


Waddell's  Iron  Hid) way  Bridges 8vo,     $4  00 

"        De  IVntibus  (a  Pocket-book  for  Bridge  Engine.  : 

> nst ruction  of  Bridges  aud  Roofs 8vo,  2  00 

Wright's  Designing  of  Draw  Spans 8vo,  2  50 

CHEMISTRY. 

QUALITATIVE—  QUANTITATIVE-    OlKiANIC— INORGANIC,   ETC. 

Adriance's  Laboratory  Calculations 12nio,  1  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Austen's  Notes  for  Chemical  Students 12mo,  1  50 

Bolton's  Student's  Guide  in  Quantitative  Analysis .8vo,  1  50 

ii's  Analysis  by  Electrolysis.    (Herrick  and  Boltwood.).8vo,  3  00 

Crafts's  Qualitative  Analysis.     (Scbaeffer.) 12ino,  1  50 

Drechsel's  Chemical  Reactions.    (Merrill.) 12mo,  1  25 

nius's  Quantitative  Chemical  Analysis.    (Allen.) 8vo,  6  00 

Qualitative          "               "            (Johnson.). .,  ..8vo,  300 

(Wells)  Trans.  16th. 

German  Edition 8vo,  5  00 

Fuerte's  Water  and  Public  Health 12mo,  1  50 

iud  Fuel  Analysis 12mo,  1  25 

Ilainmarsten's  Physiological  Chemistry.    (Mandel.) 8vo,  4  00 

Helm's  Principles  of  Mathematical  Chemistry.    ( Morgan).  12mo,  1  50 

Kolb.                             :iemistry 12mo,  1  50' 

Ladd's  Quantitative  Chemical  Analysis I2mo,  1  00 

L.-ni'                       niMiiAi'                   Single. ) 8vo,  300 

'•mical  Lalioiatory 12mo,  1   50 

\  atei'-supply 8vo,  5  00 

An:il\>i<  of  Potable  WaM  r.      (In  the  press.} 

Miller's  Clirmiral  PhjBtaa 8vo,  2  00 

^Ii\ter's  Elenieiiiary  Tr  xt-book  of  Chemistry 12ni(),  1   .">() 

Moriran's  Tin-  Thcury  <>f  Solutions  and  its  Results 12m<>.  1  00 

Nie'no!>'s  Water-supply  <(  "nemical  and  Sanitary* 8vo,  2  50 

O'llrine's  Laboratory  <  luide  to  Chemical  Analysis 8vo,  2  00 

Perkins's  Qualitative  Analysis 1'Jmo.  1   00 

Pinn-                    '••  C'hcmistry.     'Ai:-ten.) l'Jm<>.  1    :,n 

Poole's  Calorific  Power, ,f  Fuels Sv<>,  300 

I    on    Inorganic  Ciiemistry  (Xoii- 

iiietallic) OblongSvo.  moro« co.  75 

5 


RuddimaiTs  Incompatibilities  in  Prescriptions 8vo,  $2  00 

Schimpf's  Volumetric  Analysis 12mo,  2  50 

Spencer's  Sugar  Manufacturer's  Handbook .  12mo,  morocco  flaps,  2  00 
"        Handbook     for    Chemists    of   Beet     Sugar     House. 

12mo,  morocco,  3  00 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Troilius's  Chemistry  of  Iron 8vo,  2  00 

Wells's  Inorganic  Qualitative  Analysis 12mo,  1  50 

"      Laboratory  Guide  in  Qualitative  Chemical  Analysis,  8vo,  1  50 

Wiechmann's  Chemical  Lecture  Notes 12mo,  3  00 

Sugar  Analysis 8vo,  2  50 

Wulling's  Inorganic  Phar.  and  Med.  Chemistry 12mo,  2  00 

DRAWING. 

ELEMENTARY — GEOMETRICAL — TOPOGRAPHICAL. 

Hill's  Shades  and  Shadows  and  Perspective 8vo,  2  00 

MacCord's  Descriptive  Geometry 8vo,  3  00 

' '          Kinematics 8vo,  5  00 

"          Mechanical  Drawing 8vo,  4  00 

Mahan's  Industrial  Drawing.    (Thompson.) 2  vols.,  8vo,  3  50 

Reed's  Topographical  Drawing.     (II.  A.) 4to,  5  00 

Reid's  A  Course  in  Mechanical  Drawing 8vo.  2  00 

"      Mechanical  Drawing  and  Elementary  Machine  Design. 

8vo. 

Smith's  Topographical  Drawing.     (Macmillan.) 8vo,  250 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

"        Drafting  Instruments 12mo,  1  25 

' '        Free-hand  Drawing 1 2ino,  1  00 

"        Higher  Linear  Perspective  8vo,  3  50 

"        Linear  Perspective 12mo,  100 

"        Machine  Construction , 2  vols.,  8vo,  7  50 

"        Plane  Problems , 12mo,  125 

"         Primary  Geometry 12mo,  75 

Problems  and  Theorems 8vo,  2  50 

"        Projection  Drawing 12mo,  150 

"         Shades  and  Shadows 8vo,  300 

Stereotomy— Stone  Cutting 8vo,  250 

Whelpley's  Letter  Engraving 12mo,  2  00 

6 


ELECTRICITY  AND  MAGNETISM. 

1 1. 1. T. Ml  NATION  —  BATTKKIKS  —  I'lIYSK  S. 

Anthony  and  Brackett's  Text- book  of  Physics  (Magic).    . .  .8vo,  $4  00 

Barker's  Deep- sen  Soundings 8vo,  2  00 

Benjamin's  Voltaic  Cell 8vo,  3  00 

History  of  Electricity 8vo  300 

Cosmic  Law  of  Thermal  Repulsion 18mo,  75 

Crehore  and  Squier's  Experiments  with  a  New  Polarizing  Photo- 
Chronograph 8vo,  3  00 

*  Dredge's  Electric  Illuminations'.  .  .  .2  vols. ,  4to,  half  morocco,  25  00 

Vol.  II 4lo,  7  50 

Gilbert's  De  magnete.     (Mottelay .) 8vo,  2  50 

Holman's  Precision  of  Measurements 8vo,  2  00 

Michie's  Wave  Motion  Relating  to  Sound  and  Light,. 8vo,  4  00 

Morgan's  The  Theory  of  Solutions  and  its  Results 12mo,  1  00 

Ninmk-t's  Electric  Batteries.     (Fishback.) .  .12mo,  2  50 

Reagan's  Steam  and  Electrical  Locomotives 12mo,  2  00 

Thurston's  Stationary  Steam  Engines  for  Electric  Lighting  Pur- 
poses  12mo,  150 

Tillman's  Heat 8vo,  1  50 

ENGINEERING. 

CIVIL — MECHANICAL— SANITARY,  ETC. 
(See  also  BRIDGES,    p.  4  ;  HYDRAULICS,    p.  8 ;   MATERIALS  OF  EN- 

(,IM .1  -.1:1  Mi.  p.  11  ;   .Mi:<  HANICS  AND  MACHINERY,  p.  11  ;    STEAM  ENGINES 

AND  BOILERS,  p.  14.) 

Baker's  Masonry  Construction 8vo,  5  00 

Surveying  Instruments 12mo,  3  00 

Black's  U.  S.  Public  Works 4to,  5  00 

Brook's  Street  Railway  Location 12mo,  morocco,  1  50 

Butts's  Engineer's  Field-book 12mo,  morocco,  2  50 

Byrne's  Highway  Construction 8vo,  7  50 

Inspection  of  Materials  and  Workmanship.    12mo,  mor. 

Carpenter's  Experimental  Engineering  8vo,  6  00 

Church's  Mechanics  of  Engineering— Solids  and  Fluids 8vo,  6  00 

Notes  and  Examples  in  Mechanics.  .  . 8vo,  2  00 

Crandall's  Earthwork  Tables    8vo,  1  50 

The  Transition  Curve 12mo,  morocco,  1  50 

7 


*  Dredge's  Peim.  Railroad  Construction,  etc.  .  .  Folio,  half  mor.,  $20  00 

*  Drinker's  Tunnelling 4to,  half  morocco,  25  00 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

Fowler's  Coffer-dam  Process  for  Piers 8vo. 

Gerhard's  Sanitary  House  Inspection 16mo,  1  00 

Godwin's  Railroad  Engineer's  Field-book.  12mo,  pocket-bk.  form,  2  50 

Gore's  Elements  of  Geodesy , Svo,  2  50 

Howard's  Transition  Curve  Field-book 12mo,  morocco  flap,  1  50 

Howe's  Retaining  Walls  (New  Edition.) .12mo,  1  25 

Hudson's  Excavation  Tables.     Vol.  II , 8vo,  1  00 

liutton's  Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Johnson's  Materials  of  Construction Svo,  6  00 

Stadia  Reduction  Diagram.  .Sheet,  22 J  X  28£  inches,  50 

"  Theory-  and  Practice  of  Surveying 8vo,  4  00 

Kent's  Mechanical  Engineer's  Pocket-book 12mo,  morocco,  5  00 

Kiersted's  Sewage  Disposal 12mo,  1  25 

Kirkwood's  Lead  Pipe  for  Service  Pipe 8vo,  1  50 

Mahau's  Civil  Engineering.  (Wood.) 8vo,  5  00 

Merriinan  and  Brook's  Handbook  for  Surveyors. .  .  .12mo,  mor.,  2  00 

Merriman's  Geodetic  Surveying Svo,  2  00 

"  Retaining  Walls  and  Masonry  Dams Svo,  2  00 

Mosely's  Mechanical  Engineering.  (Maliaii.) Svo,  5  00 

Nagle's  Manual  for  Railroad  Engineers /.  .12mo,  morocco,  o  00 

Pattou's  Civil  Engineering ,8vo,  7  50 

"  Foundations... Svo,  500 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Ruffuer's  Non-tidal  Rivers.. Svo,  1  25 

Searles's  Field  Engineering 12mo,  morocco  flaps,  3  00 

"  Railroad  Spiral 12nio,  morocco  flaps.  1  50 

Siebert  and  Biggin's  Modern  Stone  Cutting  and  Masonry. .  .8vo,  1  50 

Smith's  Cable  Tramways 4to,  2  50 

"  Wire  Manufacture  and  Uses 4to,  3  00 

Spalding's  Roads  and  Pavements 12mo,  2  00 

Hydraulic  Cement. 12mo,  2  00 

Thurston's  Materials  of  Construction  8vo,  5  00 

*  Trautwiue's  Civil  Engineer's  Pocket-book.  ..12mo,  mor.  flaps,  5  00 

*  "           Cross-section ,  Sheet,  25 

*  "           Excavations  and  Embankments Svo,  2  00 

8 


*  Trautwiue's  Laying  Out  Curves 12mo,  morocco,  $2  50 

Waddell's  Oe  Poutibus  (A  Pocket-book  for  Bridge  Engine. 

12mo,  morocco,  3  00 

"Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  G  50 

"      Law  of  Field  Operation  in  Engineering,  etc 8vo. 

Warren's  Sterrotomy— Stone  Cutting 8vo,  2  50 

Webb  s  Engineering  Instruments 12mo,  morocco,  1  00 

Wegmanu's  Construction  of  Masonry  Dams 4to,  5  00 

Wellington's  Location  of  Railways  .       8vo,  500 

WhrrVr's  Civil  Engineering 8vo,  4  00 

WoliTs  Windmill  as  a  Prime  Mover 8vo,  3  00 

HYDRAULICS. 
WATftB-WHEtLft— WINDMILLS— SERVICE  PIPE— DRAINAGE,  ETC. 

(See  also  ENOIM.KKING,  p.  6.) 
Ba/in's  Experiments  upon  the  Contraction  of  the  Liquid  Vein 

(Trauiwine) .    8vo,  2  00 

]>ovey's  Treatise  on  Hydraulics 8vo,  4  00 

C'ollin's  Graphical  Solution  of  Hydraulic  Problems I'Jmo, 

1'8  Treatise  on  the  Winds,  Cyclones,  and  Tornadoes. .  .Svo,  4  00 

Euerte's  Water  and  Public  Health 12mo,  1  50 

GanguilleUv.  KutterVFlow  of  Water.  (Hering&  Trautwiue.).8vo,  4  00 

Ha/en's  Fill  ration  of  Public  Wtiter  Supply 8vo,  2  00 

1I«  r-chfl'.-  113  Experiments   Svo,  2  00 

Kiersted's   Sewage  I )i>pos.al 12mo,  1  25 

Kirkwood's  Lead  Pipe  fdr  Service  Pipe Svo,  1  HO 

1*1  Water  Supply Svo,  5  00 

Mrrriiuan's  TicatU-  on  Hydraulics. Svo,  4  00 

Nichols's  Water  Supply  (Chemical  and  Sanitary) Svo, 

KufTin-r's  Improvement  for  Non-tidal  Rivers Svo,  1  25 

We«;mamf>  Water  Supply  of  the  City  of  New  York 4to,  10  00 

LCh'8  Hydraulics.     (Du  l)oi>.) Svo,  5  00 

i  i Cation  En irincciini: Svo.  400 

Hydraulic  and  Placer  Mining 12mo,  2  00 

-  Windmill  Bi  a  Primr  Mover Svo,  3  00 

rory  of  Turbines Svo,  250 

MANUFACTURES. 

AMI.IM:  -  lJoii.i.i>     ExPLOfilVBB  -;AR— WATCHES  — 

\V<. 

Allen                  or  Iron  Ai                 Svo,  300 

Beaumont's  Woollen  ami  \Vo:                     'acture 1','mo,  1  50 

Bolluud'.-  Knc\cloptedia  of  Founding  Terms 12uio,  3  00 

9 


Bolland's  The  Iron  Founder 12mo,  2  50 

"        Supplement 12mo,  250 

Booth's  Clock  and  Watch  Maker's  Manual 12mo,  2  00 

Bouvier's  Handbook  on  Oil  Painting 12mo,  2  00 

Eissler's  Explosives,  Nitroglycerine  and  Dynamite 8vo,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

Metcalf 's  Steel— A  Manual  for  Steel  Users 12mo,  $2  00 

Reimaun's  Aniline  Colors.     (Crookes.) ,8vo,  2  50 

*  Reisig's  Guide  to  Piece  Dyeing 8vo,  25  00 

Spencer's  Sugar  Manufacturer's  Handbook 12mo,  inor.  flap,  2  00 

"        Handbook      for      Chemists       of      Beet       Houses. 

12mo,  mor.  flap,  3  00 

Svedelius's  Handbook  for  Charcoal  Burners 12mo,  1  50 

The  Lathe  and  Its  Uses 8vo,  6  00 

Thurston's  Manual  of  Steam  Boilers 8vo,  5  00 

Walke's  Lectures  on  Explosives 8vo,  4  00 

West's  American  Foundry  Practice 12m o,  2  50 

"      Moulder's  Text-book 12mo,  2  50 

Wiechmanu's  Sugar  Analysis 8vo,  2  50 

Woodbury 's  Fire  Protection  of  Mills 8vo,  2  50 


MATERIALS  OF  ENGINEERING. 

STRENGTH — ELASTICITY — RESISTANCE,  ETC. 
(See  also  ENGINEERING,  p.  6.) 

Baker's  Masonry  Construction 8vo,  5  00 

Beardslee  and  Kent's  Strength  of  Wrought  Iron 8vo,  1  50 

Bovey's  Strength  of  Materials 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  Materials 8vo,  5  00 

Byrne's  Highway  Construction .    v  .  .8vo,  5  00 

Carpenter's  Testing  Machines  and  Methods  of  Testing  Materials. 

Church's  Mechanics  of  Engineering — Solids  and  Fluids 8vo,  6  00 

Du  Bois's  Stresses  in  Framed  Structures 4to,  10  00 

Hatfield's  Transverse  Strains 8vo,  5  00 

Johnson's  Materials  of  Construction 8vo,  6  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Merriman's  Mechanics  of  Materials , 8vo,  4  00 

"              Strength  of  Materials. 12mo,  1  00 

Pattou's  Treatise  on  Foundations 8vo,  5  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Spaldiug's  Roads  and  Pavements 12mo,  2  00 

Thurston's  Materials  of  Construction , , ....  ,8vo,  5  00 

10 


mrston's  Materials  of  Engineering 3  vols.,  8vo,  $8  00 

Vol.  I.,  Non-metallic  8vo,  200 

Vol.  II.,  Iron  and  Steel 8vo,  3  50 

Vol.  III.,  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Weyrauch's  Strength  of  Iron  and  Steel.    (Du  Bois.) 8vo,  1  50 

Wood's  Resistance  of  Materials 8vo,  2  00 

MATHEMATICS. 

CALCULUS— GEOMETRY — TRIGONOMETRY,  ETC. 

Baker's  Elliptic  Functions 8vo,  1  50 

Ballurd's  Pyramid  Problem 8vo,  1  50 

Barnard's  Pyramid  Problem 8vo,  1  50 

Bass's  Differential  Calculus 12mo,  4  00 

Brigg's  Plane  Analytical  Geometry 12mo,  1  00 

Chapman's  Theory  of  Equations 12ino,  1  50 

Chessin's  Elements  of  the  Theory  of  Functions. 

Compton's  Logarithmic  Computations 12mo,  1  50 

Craig's  Linear  Differential  Equations 8vo,  5  00 

Da  vis's  Introduction  to  the  Logic  of  Algebra 8vo,  1  50 

Halsted's  Elements  of  Geometry c..8vo,  1  75 

Synthetic  Geometry 8vo,  150 

Johnson's  Curve  Tracing 12mo,  1  00 

Differential  Equations— Ordinary  and  Partial 8vo,  3  50 

Integral  Calculus 12mo,  1  50 

"          Unabridged. 

Least  Squares 12mo,  1  50 

flow's  Logarithmic  and  Other  Tables.     (Bass.) 8vo,  2  00 

"        Trigonometry  with  Tables.     (Bass.) 8vo,  3  00 

Mahan's  Descriptive  Geometry  (Stone  Cutting). ...  8vo,  1  50 

lUetriinan  and  Woodward's  Higher  Mathematics 8vo,  5  00 

Mi'rriman's  Method  of  Least  Squares 8vo,  2  00 

Parker's  Quadrature  of  the  Circle , 8vo,  2  50 

Rice  and  Johnson's  Differential  and  Integral  Calculus, 

2  vols.  in  1,  12mo,  2  50 

'Differential  Calculus 8vo,  3  00 

Abridgment  of  Differential  Calculus 8vo,  150 

Searles's  Elements  of  Geometry 8vo,  1  50 

Totten's  Metrology 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

' '        Drafting  Instruments 12mo,  1  25 

Free-hand  Drawing 12mo,  100 

"        Higher  Linear  Perspective 8vo,  3  50 

Linear  Perspective 12mo,  1  00 

Primary  Geometry 12mo,  75 

11 


Warren's  Plane  Problems. 12mo,  $1  25 

"  Problems  and  Theorems 8vo,  2  50 

"  Projection  Drawing 12mo,  1  50 

Wood's  Co-ordiuate  Geometry 8vo,  2  00 

"  Trigonometry 12mo,  1  00 

Woolf  s  Descriptive  Geometry Royal  8vo,  3  00 

MECHANICS-MACHINERY. 

TEXT-BOOKS  AND  PRACTICAL  WORKS. 
(See  also  ENGINEERING,  p.  6.) 

Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Benjamin's  Wrinkles  and  Recipes , 12mo,  2  00 

Carpenter's  Testing  Machines  and   Methods   of  Testing 

Materials 8vo. 

Chordal's  Letters  to  Mechanics 12mo,  2  00 

Church's  Mechanics  of  Engineering .8vo,  6  00 

"        Notes  and  Examples  in  Mechanics 8vo,  2  00 

Crehore's  Mechanics  of  the  Girder 8vo,  5  00 

Cromwell's  Belts  and  Pulleys 12mo,  1  50 

Toothed  Gearing 12mo,  1  50 

Comptou's  First  Lessons  in  Metal  Working 12mo,  1  50 

Dana's  Elementary  Mechanics 12mo,  1  50 

Dingey's  Machinery  Pattern  Making 12mo,  2  00 

Dredge's     Trans.     Exhibits     Building,      World     Exposition, 

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